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

1,2

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

Links

Table of n, a(n) for n=1..13.

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A044297 A044678 A074866 * A122513 A252687 A053019

Adjacent sequences: A229204 A229205 A229206 * A229208 A229209 A229210

Keyword

nonn,more

Author

Paolo P. Lava, Sep 16 2013

Extensions

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

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

Status

approved