A344380 Complement of A344378 in A172186.

6, 14, 38, 42, 57, 65, 70, 93, 106, 114, 118, 138, 154, 158, 182, 186, 190, 205, 210, 217, 218, 222, 266, 277, 281, 285, 309, 326, 334, 366, 381, 390, 393, 394, 397, 398, 401, 406, 434, 457, 469, 473, 478, 493, 498, 505, 518, 542, 561, 570, 581, 606, 614, 618

Offset: 1

Author

René Gy, May 16 2021

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• René Gy, When the sum of the first n consecutive even (2k>0) powers is a prime number?, Math StackExchange.

Crossrefs

Cf. A172186, A344378.

Extensions

More terms added and incorrect Mathematica program removed by Jinyuan Wang, Mar 07 2025