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

1,1

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).

Links

Table of n, a(n) for n=1..54.

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

Example

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.

Crossrefs

Cf. A172186, A344378.

Sequence in context: A053560 A229441 A119874 * A270127 A269511 A270907

Adjacent sequences: A344377 A344378 A344379 * A344381 A344382 A344383

Keyword

nonn

Author

René Gy, May 16 2021

Extensions

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

Status

approved