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 A344380 #35 Mar 19 2025 10:09:52
%S A344380 6,14,38,42,57,65,70,93,106,114,118,138,154,158,182,186,190,205,210,
%T A344380 217,218,222,266,277,281,285,309,326,334,366,381,390,393,394,397,398,
%U A344380 401,406,434,457,469,473,478,493,498,505,518,542,561,570,581,606,614,618
%N A344380 Complement of A344378 in A172186.
%C A344380 Terms belong to A172186 but not to A344378. Even though a(n)*(a(n)+1)*(2*a(n)+1) is squarefree, Sum_{j=1..a(n)} j^(2k) always has a prime divisor which is smaller than 2*a(n)+3, whatever k. For the integers m such that m*(m+1)*(2*m+1) is nonsquarefree, Sum_{j=1..m} j^(2k) always has a prime divisor which is smaller than 2*m+3, whatever k, because it is divisible by any prime p such that p^2 divides m*(m+1)*(2*m+1).
%H A344380 René Gy, <a href="https://math.stackexchange.com/q/4122583/130022">When the sum of the first n consecutive even (2k>0) powers is a prime number?</a>, Math StackExchange.
%e A344380 14 belongs to the sequence, because it is squarefree, and Sum_{j=1..14} j^(2k) is always divisible by 29 when 14 does not divide k, and when 14 divides k, it is divisible by 13 or by 7.
%Y A344380 Cf. A172186, A344378.
%K A344380 nonn
%O A344380 1,1
%A A344380 _René Gy_, May 16 2021
%E A344380 More terms added and incorrect Mathematica program removed by _Jinyuan Wang_, Mar 07 2025