A229211 Numbers k such that Sum_{j=1..k} (j*(j+1)/2 - sigma(j))^j == 0 (mod k), where sigma(j) = A000203(j) and j*(j+1)/2 - sigma(j) = A024816(j).

1, 2, 9, 78, 3205, 5589, 14153, 246123

Offset: 1

Author

Paolo P. Lava, Sep 16 2013

Keywords

• nonn
• more

Comments

Tested up to k = 50000.

Examples

			(1*2 / 2 - sigma(1))^1 + (2*3 / 2 - sigma(2))^2 + ... + (9*10 / 2 - sigma(10))^9 = 35223475538772 and 35223475538772 / 9 = 3913719504308.
		

Crossrefs

Cf. A000203, A227427, A227429, A227502, A227848, A229095, A229207-A229210.

Programs

Maple

with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+(n*(n+1)/2-sigma(n))^n; if t mod n=0 then print(n); fi; od; end: P(10^6);

PARI

isok(n) = sum(i=1, n, (i*(i+1)/2 - sigma(i))^i) % n == 0; \\ Michel Marcus, Nov 09 2014

Extensions

Typo in name and crossref corrected by Michel Marcus, Nov 09 2014

a(8) from Kevin P. Thompson, Apr 20 2022