A349989 a(n) is the smallest k such that k^n + (k+1)^n is divisible by a square > 1.
4, 3, 1, 113, 2, 3, 3, 19, 1, 1, 4, 113, 4, 3, 1, 765, 4, 3, 4, 87, 1, 3, 4, 19, 2, 2, 1, 28, 4, 1, 4, 151, 1, 3, 2, 113, 4, 3, 1, 19, 4, 3, 4, 113, 1, 3, 4, 335, 3, 1, 1, 113, 4, 3, 1, 19, 1, 3, 4, 87, 4, 3, 1, 379, 2, 3, 4, 1, 1, 1, 4, 19, 4, 3, 1, 113, 3, 1, 4
Offset: 1
Keywords
Links
- Kevin P. Thompson, Factorizations to support a(n) for n = 1..79
Programs
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PARI
a(n) = my(k=1); while(issquarefree(k^n + (k+1)^n), k++); k; \\ Michel Marcus, Dec 09 2021
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Python
from sympy.ntheory.factor_ import core def squarefree(n): return core(n, 2) == n def a(n): k = 1 while squarefree(k**n + (k+1)**n): k += 1 return k print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Dec 09 2021
Formula
a(n) = A289629(n) if n is even.
a(k) = 1 for k in A049096.
a(n) <= 4 if 4 does not divide n; among terms where 4 divides n, certain terms appear repeatedly. E.g.,
a(n) <= 113 for n == 4 (mod 8): for all such n, 17^2 divides 113^n + 114^n;
a(n) <= 19 for n == 8 (mod 16): for all such n, 17^2 divides 19^n + 20^n;
a(n) <= 765 for n == 16 (mod 32): for all such n, 97^2 divides 765^n + 766^n;
a(n) <= 87 for n == 20 (mod 40): for all such n, 41^2 divides 87^n + 88^n;
a(n) <= 28 for n == 68 (mod 136): for all such n, 17^2 divides 28^n + 29^n;
a(n) <= 151 for n == 32 (mod 64): for all such n, 257^2 divides 151^n + 152^n;
a(n) <= 335 for n == 48 (mod 96): for all such n, 769^2 divides 335^n + 336^n.
Extensions
a(64)-a(79) from Kevin P. Thompson, Feb 23 2022
Comments