A242927 a(n) = smallest k such that k^m + (k+1)^m + ... + (k+m-1)^m is prime, where m = A014117(n).

2, 1, 4, 99, 3081

Offset

1,1

Comments

Original title: "Numbers m such that k^m + (k+1)^m + ... + (k+m-1)^m is prime for some k." That sequence is a duplicate of A014117 as first noticed by M. F. Hasler. Indeed, Faulhaber's formula implies that m divides k^m + (k+1)^m + ... + (k+m-1)^m for any k, unless m divides the denominator of Bernoulli number B_m. Hence, the old sequence is a subsequence of A014117, while the discovered primes prove that the two sequences coincide. The current sequence lists the smallest values of k instead. - Max Alekseyev, Nov 04 2025

Links

Table of n, a(n) for n=1..5.

Example

For n = 1, m = A014117(1) = 1 and the sum is k^1 = k, which is prime for smallest k = 2 = a(1) or any other prime (cf. A000040).
For n = 2, m = A014117(2) = 2 and the sum is k^2 + (k+1)^2, which is prime for smallest k = 1 = a(2).
a(5) = 3081 with m = 1806 yields a strong PRP with 6663 digits. - Don Reble, Mar 23 2018

Prog

(PARI) K(n)=for(k=1, oo, if(ispseudoprime(sum(i=0, n-1, (k+i)^n)), return(k)))
foreach([1, 2, 6, 42, 1806], n, if(v=K(n), print(v)));  \\ Edited by M. F. Hasler, Mar 23 2018 and Max Alekseyev, Nov 04 2025

Crossrefs

Cf. A000040, A027861, A014117.

Sequence in context: A383083 A355517 A227050 * A093876 A375605 A322334

Adjacent sequences: A242924 A242925 A242926 * A242928 A242929 A242930

Keyword

nonn,full,fini

Author

Derek Orr, May 26 2014

Extensions

a(5) from Don Reble, Mar 23 2018

Example corrected and extended by M. F. Hasler, Apr 05 2018

Original sequence was a duplicate of A014117. Edited and repurposed to list the corresponding values of k by Max Alekseyev, Nov 04 2025

Status

approved