This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A289629 #60 Jan 10 2022 10:59:45
%S A289629 3,7,113,14,3,23,19,7,1,2,113,75,3,7,765,36,3,2476,87,1,3,165,19,14,2,
%T A289629 7,28,149,1,2972,151,2,3,14,113,977,3,5,19,34,3,135,113,7,3,136,335,
%U A289629 23,1,7,113,11,3,2,19,2,3
%N A289629 Smallest positive k such that (k+1)^n + (-k)^n is divisible by a square greater than 1.
%C A289629 From _Kevin P. Thompson_, Dec 03 2021: (Start)
%C A289629 Most of the factorizations to prove known terms of this sequence have been uploaded to factordb.com.
%C A289629 For known terms up to n = 100, the square that divides (k+1)^n + (-k)^n is very small, less than 1500^2, with only one example greater than 1000^2.
%C A289629 a(20m+10) = 1 for m >= 0. With k = 1 and starting at n = 20*0 + 10 = 10, (k+1)^n + (-k)^n = 2^10 + 1 = 1024 + 1 = 1025 which is divisible by 5^2. Since the last two digits of 2^n repeat in a cycle of length 20, (k+1)^n + (-k)^n will always be divisible by 5^2 for n = 20m + 10.
%C A289629 Conjecture: (k+1)^n + (-k)^n is not squarefree for the following (n, k) patterns, with m >= 1: (22m-11, 2), (20m-6, 3), (20m-2, 3), (3^m, 7), (15m, 7), (20m-15, 14), (16m-8, 19), (42m-35, 23), and (8m-4, 113). In each case, the value of a(n) in this sequence is usually equal to the value specified for k, but sometimes this value is not the smallest such k. For example, in the (n, k) = (20m-2, 3) case, a(20m-2) = 3 for m = 1..3, but at m = 4, a(20*4-2) = a(78) = 1.
%C A289629 (End)
%H A289629 Kevin P. Thompson, <a href="/A289629/a289629_6.txt">Table of n, a(n) for n = 2..150</a> with unconfirmed terms (term 31 confirmed by _Hugo Pfoertner_ and term 48 confirmed by _Jon E. Schoenfield_ and _Hugo Pfoertner_)
%F A289629 a(2n+1) = A280547(2n+1).
%e A289629 a(2) = 3 because (1+1)^2 + (-1)^2 = 5 is squarefree, (2+1)^2 + (-2)^2 = 13 is squarefree, and (3+1)^2 + (-3)^2 = 25 is divisible by 5^2.
%t A289629 Table[k=1;While[SquareFreeQ[(k+1)^n+(-k)^n],k++];k,{n,2,15}] (* _Giorgos Kalogeropoulos_, Dec 03 2021 *)
%o A289629 (PARI) a(n) = my(k=1); while (issquarefree((k+1)^n + (-k)^n), k++); k; \\ _Michel Marcus_, Dec 04 2021
%Y A289629 Cf. A280547, A285929.
%Y A289629 Cf. A280302, A289985.
%K A289629 nonn,more
%O A289629 2,1
%A A289629 _Juri-Stepan Gerasimov_, Sep 02 2017
%E A289629 Offset corrected; a(16), a(32), a(36), a(44), and a(48) corrected; and a(50)-a(58) added by _Kevin P. Thompson_, Dec 05 2021.