A309556 Composite numbers m such that m divides Sum_{k=1..m-1} (lcm(1,2,...,(m-1)) / k)^2.

52781, 782957, 1395353, 2602439

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Aug 07 2019

Keywords

• nonn
• bref
• more

Comments

Composites m such that m | H_2(m-1) * lcm(1^2,2^2,...,(m-1)^2), where H_2(m) = 1/1^2 + 1/2^2 + ... + 1/m^2.

By Wolstenholme's theorem, if p > 3 is a prime, then p divides the numerator of H_2(p-1) and thus H_2(p-1) * lcm(1,2^2,...,(p-1)^2) == 0 (mod p). This sequence is formed by the pseudoprimes that are solutions of this congruence.

a(5) > 10^7 if it exists.

Links

• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
• Wikipedia, Wolstenholme's theorem.

Crossrefs

Cf. A007406, A007407, A025529 (see our comment), A051418.

Programs

Mathematica

s = 0; c = 1; n = 1; seq = {}; Do[s += 1/n^2; c = LCM[c, n^2]; n++; If[CompositeQ[n] && Divisible[s*c, n], AppendTo[seq, n]], {2 * 10^6}]; seq