A229207 Numbers k such that Sum_{j=1..k} tau(j)^j == 0 (mod k), where tau(j) = A000005(j), the number of divisors of j.

1, 46, 135, 600, 1165, 1649, 5733, 6788, 6828, 9734, 29686, 363141, 1542049

Offset: 1

Author

Paolo P. Lava, Sep 16 2013

Keywords

• nonn
• more

Comments

a(12) > 200000. - Michel Marcus, Feb 25 2016

a(13) > 500000. - Harvey P. Dale, Dec 13 2018

a(14) > 3000000. - Jason Yuen, Feb 27 2024

Examples

			tau(1)^1 + tau(2)^2 + ... + tau(45)^45 + tau(46)^46 = 1^1 + 2^2 + ... + 6^45 + 4^46 = 86543618042218910328339719795268200166 and 86543618042218910328339719795268200166 / 46 = 1881383000917802398442167821636265221.
		

Crossrefs

Cf. A000005, A227427, A227429, A227502, A227848, A229095, A229208, A229209, A229210, A229211.

Programs

Maple

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

Mathematica

Module[{nn=30000,ac},ac=Accumulate[Table[DivisorSigma[0,i]^i,{i,nn}]];Select[ Thread[{ac,Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]](* Harvey P. Dale, Dec 13 2018 *)

PARI

isok(n) = sum(i=1, n, Mod(numdiv(i), n)^i) == 0; \\ Michel Marcus, Feb 25 2016

Extensions

a(12) added by Harvey P. Dale, Dec 13 2018

a(13) added by Jason Yuen, Feb 27 2024