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
Keywords
Examples
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.
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
Programs
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Mathematica
Table[k=1;While[SquareFreeQ[n^k+(n+1)^k],k++];k,{n,33}] (* Giorgos Kalogeropoulos, Dec 08 2021 *) -
PARI
a(n) = my(k=1); while(issquarefree(n^k + (n+1)^k), k++); k; \\ Michel Marcus, Dec 08 2021
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
Extensions
Thanks to Hugo Pfoertner for computing terms a(50) and a(68).
Comments