The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A347821 Smallest prime p such that n*p+1 is a perfect power, or 0 if no such p exists. 1
 3, 13, 5, 2, 3, 2801, 5, 3, 7, 50544702849929377, 13, 2, 2, 241, 13, 3, 19, 19, 17, 463, 3, 11, 89, 2, 23, 757, 29, 732541, 31, 917087137, 29, 7, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For every n, all sufficiently large primes p such that n*p+1 is a perfect power are of the form ((n+1)^q-1)/n with q prime. a(34) = (35^313-1)/34 is too large to include, it has 482 decimal digits. a(35) - a(37) = {37, 61, 1483}. a(38) = (39^349-1)/38 is too large to include, it has 554 decimal digits. a(39) - a(100) = {5, 2, 43, 3500201, 5, 71, 43, 3851, 178481, 11, 47, 3221, 5, 178250690949465223, 2971, 127, 53, 3, 7, 3541, 61, 2, 59, 2, 61, 17, 3, 751410597400064602523400427092397, 21700501, 4831, 7, 19, 73, 5, 7, 5701, 73, 6007, 79, 39449441, 6481, 19, 79, 48037081, 6218272796370530483675222621221, 2, 3, 438668366137, 89, 5, 23, 331, 89, 654022685443, 11, 1001523179, 97, 3, 792806586866086631668831, 9901, 97, 10303}. If n*p+1 = m^k, then n*p = m^k-1 = (m-1)*(m^(k-1) + m^(k-2) + ... + m + 1). If p >= n, then m^k = n*p+1 >= n^2+1 > n^2, and we have these three cases: Case 1: m-1 > n, then p can't be prime. Case 2: m-1 = n, this is A084738. Case 3: m-1 < n. If gcd(n, m-1) != m-1, then because m^(k-1) + m^(k-2) + ... + m + 1 > n, p can't be prime. This implies m-1 | n. The three cases means that we only need to check p < n and numbers m such that m-1 | n. The first n such that a(n) = 0 are {124, 215, 224, 242, ...}, a(268) is unknown, it is the smallest prime of the form (269^q-1)/268 with prime q if exists (if such prime exists, then it must be greater than (269^63659-1)/268), otherwise 0. LINKS Eric Chen, Table of n, a(n) for n = 1..300 (with unknown term a(268)). FORMULA a(n) <= A084738(n+1) if A084738(n+1) > 0. PROG (PARI) a(n)=forprime(p=2, 2^32, if(ispower(n*p+1), return(p))) (PARI) b(n)=forprime(p=2, 2^16, if(ispseudoprime(q=((n+1)^p-1)/n), return(q))) a(n)=forprime(p=2, 2^30, if(ispower(n*p+1), return(p))); b(n) \\ this program might be incorrect beyond a(300) CROSSREFS Cf. A001597, A084738. Sequence in context: A253685 A122478 A164558 * A128368 A050089 A282174 Adjacent sequences:  A347818 A347819 A347820 * A347829 A347830 A347831 KEYWORD nonn AUTHOR Eric Chen, Sep 25 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)