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November 17

nth composite number divisible by n

Are 1, 2, 5, 6, and 7 the only positive integers n for which the nth composite number is divisible by n? GeoffreyT2000 (talk) 16:14, 17 November 2018 (UTC)[reply]

Looks like it. There aren't any other small ones, then the number of composite is more than half of the upper number so it can't divide it. Bubba73 ^{You talkin' to me?} 16:58, 17 November 2018 (UTC)[reply]

Here's a proof. Please find a flaw if possible:

The nth composite number is bounded below by n.
Because all even numbers greater than 2 and some odd numbers are composite, the presence of at least 2 composite odd numbers proves that the nth composite number will always be less than 2n for all n where 2n > the second odd composite number (15.)
Therefore, for all n >= 8, the nth composite number will be between n and 2n, and dividing it by n will give a number between 1 and 2 and thus it can't be an integer.

Can anyone find a flaw here?? Georgia guy (talk) 17:51, 17 November 2018 (UTC)[reply]

I think that's right. After that point the ratio is always strictly between 1 and 2. Bubba73 ^{You talkin' to me?} 01:31, 18 November 2018 (UTC)[reply]

A related question: If a(n) is the nth composite number, are there an infinite number of n's so that a(n)/n = 1+1/k for some k? For example (assuming my calculations are correct) a(n)/n = 3/2 for n = 24, 26, 28, 30, 32 and a(n)/n = 4/3 for n = 93, 96, 99, 105. Note that pi(a(n)) = a(n)-n+1 where pi is the Prime-counting function. --RDBury (talk) 05:08, 18 November 2018 (UTC)[reply]

Do you mean for a fixed k? Off the top of my head, probably not. The primes form a less and less dense subset of the numbers, so the number of composites approaches 100%. So I think that eventually a(n)/n will be > 1+1/k for a fixed k, but I need to think about it more (later). Bubba73 ^{You talkin' to me?} 05:21, 18 November 2018 (UTC)[reply]

I meant some k where k depends on n. (For a fixed k the number of n's would be finite by essentially the same proof as given above.) Actually, by applying the Prime number theorem I believe you can show a(n)/n ~ 1 + 1/log n, which gives e^k as a ballpark estimate for n in terms of k. I did some number crunching and found that solutions exist for all k ≤ 14; for k = 14 there are 15 values of n ranging from 8921290 to 8922830. My processing power and programming skills are limited so 14 is about as high as I can go. --RDBury (talk) 12:02, 18 November 2018 (UTC)[reply]

Correction, actually a(n)/n ~ 1 + 1/log n is obvious since a(n)/n and 1 + 1/log n are both ~1. What I should have said was a(n)/n = 1 + 1/log n + o(1/log n). --RDBury (talk) 17:58, 18 November 2018 (UTC)[reply]

Using OEIS:A038625, the smallest n for k is around A038625(k+1)×k/(k+1). Assuming correctness of the OEIS sequence, the smallest for k = 1 to 49 is: 5, 24, 93, 276, 905, 2658, 7364, 21608, 58095, 159280, 440803, 1204344, 3272191, 8921290, 24257625, 65990992, 179408055, 487205532, 1324483920, 3599858180, 9781165863, 26580367572, 72229699468, 196296050304, 533467135175, 1449815874780, 3940263612747, 10709036057524, 29105918065900, 79107390087240, 215010172388833, 584394426250240, 1588390452684816, 4317301265437368, 11734671691292690, 31895736939042660, 86695604114932233, 235648080307087472, 640520326622683767, 1741020550592597280, 4732347545674762806, 12863257280167193676, 34964451581289156721, 95039423137755548716, 258334285328363071485, 702200908598678530374, 1908717748466836346374, 5188274437378776764592, 14102788619438915542083. My value for k = 14 agrees with RDBury. PrimeHunter (talk) 18:17, 18 November 2018 (UTC)[reply]

I think I see what you're doing, so let me do an example and tell me know if it's what you had in mind. From OEIS you have pi(330) = 66. So taking a(n) = 330 in the equation pi(a(n))=a(n)-n-1 (I had a sign wrong above), and solving for n you get a(263)=330. Define f(n) = n-4(a(n)-n) = 5n-4a(n), so f(263)=-5. If n is increased by 1 then f(n) either increases by 1 or decreases (by 3), also f→∞ as n→∞, so f must eventually be 0 and since you're starting so close to 0 the computation to get to find the exact value where this happens is feasible. In fact you can iterate n→n-f(n)=4(a(n)-n) until you reach a fixed point, in this case 276 after 3 iterations. I'm guessing that Golomb's proof that the entries in A038625 always exist follows the same sort of logic, and it would imply that A038625(k) is always < the value of n we're looking for. In any case, the answer to the question posed above is yes, in fact for every k there is n so that a(n)/n = 1+1/k. It may also be interesting to note that the ratio of successive entries in your sequence seems to converge to e as you might expect. --RDBury (talk) 08:28, 19 November 2018 (UTC)[reply]

Yes, that was basically my algorithm. After reading the terms of A038625, a PARI/GP line computed the list in 0.2s. PrimeHunter (talk) 11:48, 19 November 2018 (UTC)[reply]