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A344378 Positive integers m for which there exists a positive even integer 2k such that Sum_{j=1..m} j^(2k) has no prime divisor smaller than 2*m + 3.

%I A344378 #44 Apr 16 2025 10:29:24
%S A344378 1,2,5,10,21,29,30,33,34,41,46,61,66,69,77,78,82,86,101,102,105,109,
%T A344378 110,113,129,133,141,142,145,165,173,177,178,185,194,201,209,213,214,
%U A344378 221,226,230,246,254,257,258,273,282,286,290,298,313,317,321,322,329,330
%N A344378 Positive integers m for which there exists a positive even integer 2k such that Sum_{j=1..m} j^(2k) has no prime divisor smaller than 2*m + 3.
%C A344378 a(n)*(a(n)+1)*(2a(n)+1) must be squarefree, so this is a subsequence of A172186. It is the complement of A344380 in A172186.
%C A344378 Let L= LCM_j[(p_j-1)/2], where p_j run through the set of the prime divisors of  a(n)*(a(n)+1)*(2a(n)+1). For a given member a(n) any admissible k must be a multiple of L, and for any prime p smaller than a(n) such that (p-1)/2 divides L, it holds that p does not divide a(n)-Floor[a(n)/p]. But the converse is not true: 397 is squarefree and satisfies the former condition, but Sum_{j=1..397} j^(2k) is always divisible either by 17 or by 73. 397 is the smallest "false positive" with the above test. Other "false positives" are rather scarce: 397,469,478,561,885,1002,1554,1658,1702,1977,... - _René Gy_, Apr 15 2025
%H A344378 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 A344378 2 belongs to the sequence since 1 + 2^(2*2) = 17 is a prime number which is larger than 2*2 + 1 = 5.
%e A344378 5 belongs to the sequence because 1 + 2^20 + 3^20 + 4^20 + 5^20 = 96470431101379 = 137*704163730667 has no prime divisor smaller than 2*5 + 3 = 13.
%Y A344378 Cf. A172186, A344380.
%K A344378 nonn
%O A344378 1,2
%A A344378 _René Gy_, May 16 2021
%E A344378 Incorrect Mathematica program removed by _Jinyuan Wang_, Mar 18 2025