A349988 a(n) is the smallest k such that n^k + (n+1)^k is divisible by a square > 1.

3, 5, 2, 1, 11, 10, 3, 10, 19, 3, 10, 1, 1, 29, 26, 3, 5, 3, 3, 2, 2, 1, 10, 1, 3, 10, 5, 2, 9, 3, 1, 5, 10, 3, 39, 10, 1, 7, 21, 1, 5, 5, 3, 21, 7, 2, 5, 10, 1, 78, 10, 3, 2, 26, 3, 10, 5, 1, 7, 1, 3, 1, 10, 3, 21, 7, 1, 3, 68, 3, 2, 5, 1, 21, 26, 1, 5, 2, 3

Offset

1,1

Comments

a(285) <= 111. - Kevin P. Thompson, Jan 13 2022

Links

Kevin P. Thompson, Table of n, a(n) for n = 1..284

Kevin P. Thompson, Factorizations to support a(n) for n = 1..284

Formula

a(9m-5) = 1 for m >= 1 since a(9m-5) = (9m-5)^1 + (9m-5+1)^1 = 18m-9 which is divisible by 9 = 3^2. - Kevin P. Thompson, Jan 13 2022

a(n) = 1 if n is in A046092; in this case, if n = A046092(m), then n^1 + (n+1)^1 = (2*m+1)^2. - Bernard Schott, Jan 17 2022

Example

a(2) = 5 since the values of 2^k + (2+1)^k for k=1..4 are 5, 13, 35, and 97, none of which are divisible by a square > 1, and 2^5 + (2+1)^5 = 275 which is divisible by 25 = 5^2.

Mathematica

Table[k=1; While[SquareFreeQ[n^k+(n+1)^k], k++]; k, {n, 33}] (* Giorgos Kalogeropoulos, Dec 08 2021 *)

Prog

(PARI) a(n) = my(k=1); while(issquarefree(n^k + (n+1)^k), k++); k; \\ Michel Marcus, Dec 08 2021

Crossrefs

Cf. A046092, A280302, A280547, A289629, A289985, A349989.

Sequence in context: A126353 A094791 A243524 * A272300 A115406 A059246

Adjacent sequences: A349985 A349986 A349987 * A349989 A349990 A349991

Keyword

nonn

Author

Jon E. Schoenfield, Dec 07 2021

Extensions

Thanks to Hugo Pfoertner for computing terms a(50) and a(68).

Status

approved