Pete's Blog

The following are the first twenty terms of a sequence:

0, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

If it hasn't been solved by Thursday, I'll give a hint. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

I'll post the solution on next week's sequence.

Last week's sequence (5, 13, 17, 25, 29, 37, 41, 53, 61, 65, . . .) was the set of hypotenuses of primitive Pythagorean Triples. Suppose you have integers a, b and c. They form a primitive Pythagorean Triple if 1. a²+b²=c², and 2. a and b coprime (i.e., they share no prime factors, i.e., their greatest common denominator is 1). For the triple a, b and c that satisfies the previous equation, we say that c is the hypotenuse because it is the longest side, and it is opposite the right angle. E.g., 5 is in the sequence because 5²=3²+4², and 3 and 4 share no prime factors. E.g., 37²=12²+35², and 12 and 35 share no prime factors.

Alternatively, this is the list of terms from the sequence two weeks ago (0, 1, 2, 4, 5, 8, 9, 10, 13, 16, . . .) with some excluded. Suppose we have a term from two weeks ago, say a(i) for some i=1, 2, 3, . . ., where a(i)= c²+d² for some integers c and d. Then a(i) is excluded from last week's sequence if and only if c and d share a prime factor or c and d are both odd or one of c and d is 0. I mentioned in the hint last week that Euclid's formula generates these terms. Suppose you have odd integer c and even integer d who share no prime factor. These generate the primitive Pythagorean Triple 2cd, |c²-d²| and c²+d². E.g., 0=0²+0², c=d=0. But 0 and 0 share an infinite number of prime factors, so 0 was not in last week's sequence.

Practically speaking, the exclusion statements in the previous paragraph exclude three categories of terms in the sequence from two weeks ago (this is not an exhaustive list):

1. The term is a square, say k² for some integer k. Then k²=k²+0². Obviously 0's in this construction, so all terms that are squares are excluded.
   
2. The term is a square doubled, say 2k² for some k=2, 3, 4, . . . . Then 2k²=k²+k². Obviously k shares at least one prime factor with itself so all the double squares are excluded.
   
3. The term is an hypotenuse of a Pythagorean Triple, but it is not of a primitive Pythagorean Triple. A Pythagorean Triple has the form (ma) ²+(mb) ²=(mc) ² where a, b and c are integers greater than 1, and a and b share no prime factors. The primitive case is for m=1. (An example of a Pythagorean Triple than is not primitive is 45. 45²=27²+36². But 27 and 36 share the prime factor 3, so 45 is a Pythagorean Triple, but is not primitive.) These are the toughest to identify for the purposes of excluding.
   

So, e.g., exclude 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, the rest of the squares, as they fit in the first category.

E.g, exclude 8, 18, 32, 50, the rest of the double squares, as they fit in the second category.

E.g., exclude 10, 20, 26, 34 the rest of the non-primitive Pythagorean Triples, as they fit in the third category.

Real examples: E.g., 2=1²+1². Obviously 1 and 1 are both odd, so 2 is excluded. E.g., 5=1²+2². Considering 1 and 2, they share no prime factors, 2 is even so they're not both odd, and neither number is 0, so there's no reason to exclude 5. Thus, 5 is the first candidate to go into last week's sequence. E.g., 13=2²+3². Considering 2 and 3, they share no prime factors, 2 is even so they're not both odd, and neither number is 0, so there's no reason to exclude 13. It is the second candidate to go into last week's sequence.

It was not solved.

I've noticed something recently about God's behaviour that I'm going to try to remember when I pray. Specifically, it has to do with the way God changes people's minds. Let me start with two biblical examples:

Acts 9:1-18 relates the story of Paul's conversion. I find it interesting that God doesn't treat Paul like a robot, flick a switch and cause him to believe in Jesus, but Jesus confronts Paul, proving that He is indeed resurrected (cf. 1Co. 15:8, "he appeared also to me"). God chose to work through Paul's reason to convert him. I'm not saying that it was purely natural. I'm certain God supernaturally opened Paul's heart to receive the Truth, but that aside, God worked through a normal means to change Paul's mind.

(The miraculous is never guaranteed to convince people of Truth. The Ten Plagues in Egypt is such a good example. But Jesus said it well Himself, "'If they do not hear Moses and the Prophets, neither will they be convinced if someone should rise from the dead" [Lk. 16:31].)

The second example comes from Genesis 20. Position yourself in this story as though you're one of the people serving Abimelech, the king of Gerar. He takes in Sarah as his wife, and then soon after that, gives her to Abraham.

(I read this passage as implying that it was a quick turn around between the time Sarah was taken away from Abraham to when he got her back. I suspect this because Abimelech and her hadn't consummated their marriage yet. Yes, God did say that He prevented it from happening, but He did it in such a way that Abimelech took credit for it, where, I think if it had been a lot of time, even more than a few days, Abimelech would have known it wasn't his own doing.)

If I were one of the servants, I'd wonder what changed his mind. But God again didn't unilaterally cause Abimelech to reject Sarah, but He reasoned with Abimelech, "This is Abraham's wife." Somehow Abimelech knew that it was God speaking, and he knew that God was trustworthy, so he decided not to keep Sarah as his wife.

The application I'm going to make after observing this is that when I'm praying for another person's decision (e.g., for so-and-so to like me back, or for Obama to change his mind on abortion, or for my father to consider the veracity of the Gospel) I'm going to pray that God highlights things in their minds that they already know (e.g., "Emphasize for him that the Bible is reliable," "Remind President Obama that the only differences between a baby in the womb and a born baby are arbitrary for determining personhood").

I'm not in anyway going back on my Calvinism, I'm see more clearly now, though, that God works through means, and that I want to pray for Him to act in ways that He's acted before.

--

I only made this connection today, but I really wanted to share it. I spent some time trying to figure out whether there were examples where God did, figuratively speaking, flick a switch and cause a person to change his mind. I couldn't think of any examples.

That said, I don't think it's wrong to pray for someone to make a 180˚ decision without praying for a means to work through. I know that God can still create ex-nihilo ("out of nothing," typically associated with our doctrine of creation), and that He can cause someone to change his mind or make a decision in a certain direction.

The following are the first fifty terms of a sequence:

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

This is a tough one. So I'll start it off with a hint. You might have noticed this. This sequence is a subsequence of last week's sequence. Furthermore, for any i=1, 2, 3, . . ., [a(i)]² is in the sequence as well. The sequence is a list, not a function, so the index is not related to its corresponding term. Solving the pattern will involve saying why the absent terms from last week's sequence. Additionally, you might recognize the significance of this sequence, apart from figuring why there are fewer terms than last week's sequence. If that's the case, it's fine, just state the significance of the numbers and that counts as solving it.

I'll post the solution on next week's sequence.

Last week's sequence (0, 1, 2, 4, 5, 8, 9, 10, 13, 16, . . .) was the list of integers who are the sum of two squares. E.g., 10=1²+3². E.g., 16=0²+4². E.g., 25=3²+4². It was not solved.

Update: Hint: There are two ways of finding the pattern of the sequence as I said above. So I'll give hints for each direction. The first hint is for if you're trying to solve it by finding out why the members of last week's sequence are excluded: Denote last week's sequence {a(n)} and this week's sequence {b(m)}. For each i=1, 2, 3, . . ., a(i)=c²+d² for some nonnegative integers c and d. Whether or not a(i) is an element of {b(m)} is determined by how c and d relate to one another. (Don't let the case where c=1 fool you!)

From the other direction, if you're trying to find the pattern of the sequence by recognizing the significance of the terms: Euclid's formula generates these numbers. Another hint is to suggest to look at the properties of the square of each term in this sequence and the square of each term in last week's sequence that is not in this list. You might be able to identify that way what the significance is of these numbers.

In the Bible there are many times where dead people are made alive (e.g., the son of the widow of Zarephath [1K. 17:17, 19-22], Lazarus [Jn. 11], some saved Jews [Mt. 27:52, 53], Tabitha [Ac. 9:36-37, 40]). I wonder how they dealt with being taken from the presence of God. I wonder whether this has any implications on what is in the intermediate state before the Judgment.

On the bus yesterday I listened to a talk called The Pathway to Eldership: The Qualities and Making of a Leader, given by Richard Mayhue. He's the Dean of MacArthur's college and seminary, The Master's College and Seminary, respectively. I think he's an amazing speaker. Very talented.

I usually don't recommend talks or sermons that I can't link to, but this talk was so valuable that I will direct you to it even though I can't hyperlink you to it: Go to the "Media Vault" on the website of either of MacArthur's major conferences Resolved or The Shepherds' Conference. From there, if you search "Pathway to Eldership" it should be the first option. Otherwise, you can select in the first box "Shepherds' Conference," then in the second box, "Audio Downloads," "2001" in the third box, then finally "Seminar Sessions" in the fourth box. The tenth listed talk should be the one I'm pointing you to.

Listen to it. While listening to it, he started talking about how ministries get leaders who they later regret taking in. I've thought about this along similar lines before, namely, how easy it is to ask an administratively competent person who was spiritually weak to join leadership, only later to regret it later.

Dr. Mayhue in this talk addresses a different mistake that happens in calling leaders, though it's along a similar line as I have thought before. So, for those of you who will listen to the talk, maybe this will make you more interested in it. And for those who weren't going to listen to it, hopefully you'll find the following segment taken from the talk insightful, and take it as a good warning. The segment starts at 9:36.

And I asked the question, "How in the world did he ever get into the leadership at that church?" And some of you are asking the question, "How do I inherit the leaders that I have or leaders that I have had in the past?" The basic answer to that is that leadership was based on strong natural leadership rather than on strong spiritual leadership.

And if you read much leadership literature whether it's biblical or not for the most part you're reading about strong natural leadership, you're reading about men who are highly motivated, men who have all of the externals—their style, their sophistication, they're suave, . . . [he start's this list item with "they're" again, but I can't figure out what the word is, it sounds like "high why" or "high wide" but I don't know what word he's really referring to, if you download the talk and figure out the word, please comment and let me know], and handsome, they're very, very knowledgeable, they're very, very bright men, they're successful in their field, and they're immediately available to step in and solve whatever problems are there. And that's just what you've been looking for, right? . . . [I've cut out this part because I don't want you to think that I have hidden motives behind posting this section from the talk. If you're curious what I've cut, listen to the talk.]

But if you're the one selecting leadership and now have problems with them, there's a high probability that the mistake that you made was looking at his external successes rather than asking was he a spiritual or a godly man and qualified about who he was on the inside, rather than who he was on the outside.

The following are the first twenty-five terms of a sequence:

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

I'm not sure how tough this one will be. I'm running out of ideas for good ones to post. If it hasn't been solved by Thursday, I'll give a hint then. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

I'll post the solution on next week's sequence.

Last week's sequence (4, 6, 9, 10, 14, 15, 21, 25, 35, 49, . . .) was the list of composite numbers with two prime factors of equal digit length. E.g., a(7)=21, since 21=(3)(7), and each of 3 and 7 are only one digit long. That's why there's the gap between a(10)=49 and a(11)=121. KWTsui pretty-much solved it.

Update: Sorry for being so late in bringing you the hint. Hint: You'd come back again and again to this list when dealing with the Pythagorean Theorem.

Recently the Southern Baptist Convention's "Executive Committee voted to end cooperative relationship with Broadway [Baptist Church] over church's failure to clarify or correct stand." Dr. Mohler tweeted that he was in the meeting at 2:20 this afternoon, tweeted the announcement at 3:20.

Searching the Baptist Press (the Southern Baptists' newspaper), the earliest mention I could find of the issue was from December 4^th, 2007. Apparently their pastor (who'd started in April 2001) wanted to allow homosexual members to have a single picture of their couples to appear in the members' directory.

Reading a little later, another article came out March 6^th, 2008, stating that there was a vote on the issue (it is a Baptist church), and the vote passed (with less than a third of the membership voting—boo-urns). (Update: I misunderstood the vote. They took out any idea of having couples together. My current understanding is that they put pictures of the members doing ministry together, which might have the couple together, but not necessarily.) But this vote led to a petition for the pastor to be removed from the church. The vote was on March 9^th and the congregation voted to keep their pastor (although, you can hardly say that the congregation voted to keep their pastor if only half the membership voted).

The pastor ultimately resigned anyway to take a teaching post at a theological school in Georgia.

The Southern Baptist Convention's Executive Committee a year ago started considering what to do with Broadway since the denomination is confessionally opposed to homosexuality: "In the spirit of Christ, Christians should oppose . . . homosexuality." Finally they've decided. I think they made the right call.

You might find this boring since it doesn't really impact any of us. Still, I wanted to post it because the SBC's (Southern Baptist Convention) always had a special place in my heart. They started off strong, swung liberal, and then in the late 70s and 80s regained the orthodoxy the denomination was founded on. Also, they're real Baptists, honouring both credobaptism and congregationalism, plus they're complementarian. A few other things in the mix are the presence of Drs. Albert Mohler and Mark Dever, both are Southern Baptists, and the International Mission Board which works really hard to send anyone who has a desire and is qualified to go into the mission field. They really want to tell the world about Jesus. So I get pretty excited when I hear about them. Especially when it's news of them doing the right thing.

In a talk last year Dr. Mohler said that he was uncertain what direction the denomination was going to take when it came to congregations that were walking in a direction opposed to their confession. I wonder whether he had this issue in mind when he mentioned it. I'm glad to know that they had the courage to separate from people who were in clear opposition, not only to the confession, but to the Scriptures. It gives me hopes that this cooperative organization (when I called the SBC a denomination I say it in the very loosest sense of the word, since the SBC is committed to congregationalism and local church autonomy) might stand for some good years ahead.

Update: Read the Baptist Press'
article.

--

Another thing that interested me in the course of this process, was the number of members who participated in the votes, compared with the total membership. Membership inflation has been a problem in the SBC. For example, officially, there are 16.5 million members on their rolls. But Dr. Tom Ascol, pastor of an SBC church in Cape Coral Florida, said that of the 16.5 million members, only 6.1 million of them regularly attend a Sunday worship service. And we Baptists complain about the paedobaptists having unsaved members.

Sorry Dan. I haven't posted anything of substance because I don't have too much to say. I still don't.

• Did you know that the same country where Calvinism's TULIP acronym was thought up has the tulip as its national flower?
  
• I appreciate that Dr. MacArthur's Question and Answer page has a feature that allows you to report typos. I would like it if more websites had it.
  
• Saturday Dr. Mohler Tweeted that he was picking out a card for his wife, even though her birthday wasn't until Sunday. Seems strange; I wouldn't want to alert my wife (who might be reading my online material) to what I was doing in advance for her birthday.
  
• I have never heard Piper use so much humour before in a message. Very interesting and engaging autobiographical talk.
  
• I wonder whether people who enjoy speaking on the phone for endless hours have difficulty reading nonverbal communication.
  
• If I were a postmodernist I'd never say the self-defeating statement, "There is no absolute truth." I'd say, "There is only one absolute truth."
  
• I'm often surprised at how few people know this worship song. I think it's great.

The following are the first twenty terms of a sequence:

4, 6, 9, 10, 14, 15, 21, 25, 35, 49, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

If it hasn't been solved by Thursday, I'll give a hint. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

I'll post the solution on next week's sequence.

Solutions to the sequences of the previous post:

1. The Tribonacci sequence. Each term is the sum of the previous three terms. E.g., a(3)=1, a(4)=1, a(5)=2, so a(6)=4. Jer solved it.
   
2. The sequence of primes who were the sum of two primes. E.g., 2 and 17 are prime so 19=2+17 is in the sequence, i.e., a(4)=19. It was not solved.
   
3. The sequence of non-square free numbers. That is, numbers who have at least one prime factor with multiplicity. E.g., 21=(3)(7) has no prime factors with multiplicity so it is not in the sequence. E.g., 18=(2)(3²) has the prime factor 3 with a multiplicity of 2. So 18 is in the sequence, i.e., a(6)=18. It was not solved.
   
4. I'll repeat that this was not an easy one. The term a(n) was the largest exponent such that 2^a(n) divided 2n. E.g., take the index n=5. Then 2n=10. 2¹ divides 10, but 2² does not. So, a(5)=1. It was not solved.
   
5. First, define a(1)=1. For the rest of the sequence a(n) was the smallest positive integer that hadn't been used yet which shared a prime factor with a(n-1). E.g., consider n=7. Since a(6)=9, a(7) must be a multiple of 3, since the only prime factor of 9 is 3. Is a(7)=3? No, because 3 has already been used by a(5). Is a(7)=6? No, because 6 has already been used by a(3). Obviously a(7)≠9 since that's what a(6) is. Is a(7)=12? Since 12 hasn't appeared in the sequence yet, yes, a(7)=12. It was not solved.
   
6. Of the seven sequences, this was by far the most interesting. Define a(1)=1. The hint gave away most of the solution, that is, "i is in the sequence if and only if a(i) is odd." The only things left to say were to make clear the definition of the first term, and to say that the sequence is strictly increasing, that is to say, a(i+1)>a(i). E.g., a(2) is unquestionably the most fun to figure out why it is 4, so I'll leave that for the reader's pleasure. 3 is not in the sequence (since only 1 and 4 are, thus excluding 2 and 3), so a(3) must be even. The next lowest even number is 6. So a(3)=6. E.g., 4 is in the sequence so a(4) is odd. So we take the lowest odd number that greater than 6. Thus, a(4)=7.
   
7. The sequence of balanced primes. The sequence is formed by taking the mean average of each pair of primes (i.e., for all distinct pairs of primes p and q we find (p+q)/2) and if the value is prime it is kept in the sequence. Obviously the sequence is ordered least to greatest. E.g., the mean value of the primes 3 and 7 is 5 (i.e., (3+7)/2=5), and since 5 is prime, 5 is in the sequence, i.e., a(1)=5. As you can see, there aren't a lot of primes that are the mean of two other primes.
   

Proof of the exercise given for the sequence of Monday, April 14 (1, 2, 4, 6, 16, 12, 64, 24, 36, 48). I asked you to prove that for each positive integer i there existed a number with exactly i factors. It's fairly simple once you get the first step. The number 2^i-1 has exactly i distinct factors. So, there might be a number less than 2^i-1 with i factors (e.g., 48 is significantly less than 512, but they each have exactly 10 distinct factors), but there always is a number with i factors.

The following are the first ten terms of a sequence:

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

If it hasn't been solved by Wednesday, I'll give a hint. It's pretty easy, so you probably won't need a hint. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

Update: Hint: It's related to powers of 2.

I'll post the solution on Thursday. (Since I forgot to put the hint up for last week's sequence until today, I won't put the solution up for it until Wednesday of this week.)

Update: The sequence is the highest exponent for which 2 to the exponent of the entry divides the index. E.g., 2¹=2 divides 3 but 2²=4 does not, so a(3)=1. E.g., 2³=8 divides 8 but 2⁴=16 does not, so a(8)=3. E.g., 2⁰=1 divides 7 but 2 does not, so a(7)=0. E.g., 2¹=2 divides 10 but 2²=4 does not, so a(10)=1. It was not solved.

Update: Last week's second sequence (1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, . . .) was the list of smallest integers with exactly the index's number of proper divisors, that is, a(n) was the smallest positive integer with exactly n divisors. E.g., 1, 2, 3, 4, 6, 9, 12, 18, 36 all divide 36, and they are the only divisors of 36. Furthermore, no positive integer less than 36 has exactly nine divisors. So a(9)=36. It was not solved. As an extra exercise, prove that for each i=1, 2, 3, . . . there exists an integer with exactly i divisors.

The following is a list of sequences (of varying difficulty, where the difficulty is reported based on the number of terms listed, on my subjective assessment) for the following seven weeks, each with a varying number of terms listed:

1. 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, . . .
   
2. 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, . . . (Update: Hint: Suppose p and q are primes. Which for which values of p and q is p + q in the sequence?)
   
3. 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, . . . (Update: Hint: Look at the prime factors.)
   
4. 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, . . . (Update: Hint: (Upon further thinking, I should have given more terms to indicate that this sequence was tough.) It's something to do with 2^a(n).)
   
5. 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, . . . (Update: Hint: First, notice that the greatest common divisor for successive terms beyond the first (i.e., gcd(a(i), a(i + 1))>1 for each i є {2, 3, 4, . . .}. Second, this sequence contains every positive integer.)
   
6. 1, 4, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, . . . (Update: Hint: Notice that i is in the sequence if and only if a(i) is odd.)
   
7. 5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, . . . (Update: Hint: This sequence is also a subset of the primes. It's somewhat similar to the sequence five above. For each positive integer i, a(i) is related to two primes, p and q, and more specifically the sum p + q. But, this sequence is dissimilar from the one above in that this one's terms, while all are directly related to p + q they're not necessarily of the form p + q. The terms are of the form p + q with something more done to them.)
   

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

Within eight weeks from now, for any of the sequences that are still left unsolved I'll post a hint, and then on June 15^th, with that week's sequence, post the solutions.

I think that it's easy to think that the healings in the Bible are always related to physical ailments. For example, Jesus healed blindness (Jn. 9:1-7), a withered hand (Lk. 6:6-10), a leper (Mk. 1:40-42), paralysis (Mt. 9:2-7), and many other conditions. As a result, it's easy (at least for me) to think that prayers for healing should be limited to physical infirmities. But consider Psalm 107:17-20:

If I hadn't read it, I would have never guessed that the Bible had anything to say about anorexia. I'm happy that I read it, because it reminds me once again that God does care. Not only does God care when a person is poor, hungry, fatherless, physically broken, persecuted, etc., but He also cares when our minds aren't working properly. It gives me hope that God is willing to heal struggles with gluttony or depression or paranoia, or other psychological problems.

As a note of criticism of much of Christian counseling, this verse clearly says that it's sin that's responsible for this malady, not a deficient childhood where you didn't learn the lessons you needed to. (I'm very pro-biblical counseling, where Christian counseling seems semi-Pelagian at best.)

This week's was solved really quickly, so I'm posting another. The following are the first twenty terms of a sequence:

1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144, 120, 65536, 180, 262144, 240, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

If it hasn't been solved by Thursday, I'll give a hint. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

I'll post the solution on next week's sequence.

This afternoon's sequence (1, 4, 8, 9, 16, 25, 27, 32, 36, 49, . . .) was the sequence of Powerful Numbers. That is to say, the list of positive numbers where if the number has a prime divisor, the square of that prime divides it. E.g., 2 divides 8, so, since 8 is in the list it follows that 2²=4 divides 8, which is obviously true. T solved it.

Update: I've added another ten terms to the sequence so that there are now twenty.

Update: Hint: There's a relationship between n and the divisors of a(n), but probably not the category of relationship that first comes to mind. Since I put the hint up late, I'm not going to put the solution on next week's sequence (that of April 20^th) until Wednesday, April 22^nd.

I love John Piper's ministry. I've referred to it here before, and I have the sidebar on the right of my blog devoted to the resources of his that I've read or listened to lately. Yesterday I listened to a talk that he gave at Driscoll's pastors conference last year. It was the first time in a long time (a few months) that I listened to something from Piper. It filled me with conviction of sin, and a joy in the glory and holiness of God. As with most of the audio I link to in my entries, I highly recommend it.

The talk reminded me of his book The Supremacy of God in Preaching (2003). The book was probably one of the three most important books that I've read in my Christian walk (alongside Desiring God [Piper] and Evangelism and the Sovereignty of God [Packer]).

Here's a few paragraphs from the first chapter (pp. 26-27):

At 109 pages, it's a short read with fairly large print and fairly small pages. It's Piper at his best. Adam said that it's the most readable of Piper's books. I completely agree. Get it. Read it.

One of the blurbs on its back reads, "Here's a book that every preacher should read at least once a year. This book is a powerful antidote to the unbalance, self-centered preaching of today."

The following are the first twenty terms of a sequence:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

If it hasn't been solved by Thursday, I'll give a hint. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

I'll post the solution on next week's sequence.

Last week's sequence (1, 2, 3, 4, 5, 3, 7, 4, 6, 5, . . .) was the smallest number, say m, such that n divided m! (m-factorial). E.g., 10 divides 5!=120, but not 4!=24, so a(10)=5. E.g., 16 divides 6!=720, but not 5!=120, so a(16)=6. It was not solved.

The following are the first twenty terms of a sequence:

1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, 4, 13, 7, 5, 6, 17, 6, 19, 5, 7, 11, 23, 4, 10, 13, 9, 7, 29, 5, 31, 8, 11, 17, 7, 6, 37, 19, 13, 5, . . .

Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments.

If it hasn't been solved by Thursday, I'll give a hint then. Also, if you ask a polar question (i.e., a yes/no question), I will probably answer it if it relates to this sequence.

I'll post the solution on next week's sequence.

Last week's sequence (1, 2, 3, 1, 5, 6, 7, 2, 1, 10, . . .) was what's called the square-free part of n. That is to say, a(n) is the smallest positive (i.e., nonzero) divisor of n such that na(n) is a square. E.g., 18=2∙3² so (2)(18)= 2²∙3²=(2∙3)², yielding a(18)=2. KWTsui solved it.

Update: I've added another 20 terms to the sequence. Furthermore, here's a very strong hint: The sequence describes a certain relationship between n and a(n)! (i.e., a(n) factorial).