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.

1, 2, 5, 10, 21, 29, 30, 33, 34, 41, 46, 61, 66, 69, 77, 78, 82, 86, 101, 102, 105, 109, 110, 113, 129, 133, 141, 142, 145, 165, 173, 177, 178, 185, 194, 201, 209, 213, 214, 221, 226, 230, 246, 254, 257, 258, 273, 282, 286, 290, 298, 313, 317, 321, 322, 329, 330

Offset

1,2

Comments

a(n)*(a(n)+1)*(2a(n)+1) must be squarefree, so A344378 is a subsequence of A172186. A344378 is the complement of A344380 in A172186.

Links

Table of n, a(n) for n=1..57.

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

Example

2 belongs to the sequence since 1 + 2^(2*2) = 17 is a prime number which is larger than 2*2 + 1 = 5.
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.

Mathematica

lim = 330; listu = {};
listn = Select[Range[1, lim],
  SquareFreeQ[# (# + 1) (2 # + 1)] &]; listL = {};
Do[M = (Transpose[FactorInteger[m (m + 1) (2 m + 1)]][[1]] - 1)/2;
L = 1; Do[L = LCM[L, j], {j, M}];
AppendTo[listL, L], {m, listn}]; list = Transpose[{listn, listL}];
Do[n = l[[1]]; L = l[[2]];
listp = Select[Range[n-1],
   PrimeQ[#] && Mod[L, (# - 1)/2] == 0 &]; lp = Length[listp];
j = 1; While[j \[LessSlantEqual] lp, p = listp[[j]];
  If[Mod[n - Floor[n/p], p] == 0, j = lp + 2, j = j + 1]];
If[j != lp + 2, AppendTo[listu, n]];
, {l, list}]; listu

Crossrefs

Cf. A172186, A344378.

Sequence in context: A051109 A124146 A262408 * A032468 A327286 A215925

Adjacent sequences: A344375 A344376 A344377 * A344379 A344380 A344381

Keyword

nonn

Author

René Gy, May 16 2021

Status

approved