A229095 Numbers k such that Sum_{i=1..k} i^tau(i) == 0 (mod k), where tau(i) = A000005(i), the number of divisors of i.

1, 8, 9, 67, 72, 467, 801, 1071, 5141, 7193, 25688, 68488, 97768, 111816, 381061, 7829505, 17079937, 25615576, 44582211, 91110856, 639359784, 3492789629

Offset: 1

Author

Paolo P. Lava, Sep 13 2013

Keywords

• nonn
• more

Examples

			1^tau(1) + 2^tau(2) + ... + 8^tau(8) + 9^tau(9) = 1^1 + 2^2 + 3^2 + 4^3 + 5^2 + 6^4 + 7^2 + 8^4 + 9^3 = 6273 and 6273 / 9 = 697.
		

Crossrefs

Cf. A000005, A227427, A227429, A227502, A227848.

Programs

Maple

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

PARI

list(lim) = {my(s = 0, f); for(k = 1, lim, s += k^numdiv(k); if(!(s % k), print1(k, ", ")));} \\ Amiram Eldar, Dec 29 2024

Extensions

a(16)-a(18) from Jinyuan Wang, Feb 18 2021

a(19)-a(22) from Amiram Eldar, Dec 29 2024