A068978 Numbers k such that Sum_{d|k} tau(d)/d is an integer, where tau(x) = A000005(x).

1, 2, 9, 18, 105, 210, 24375, 48750, 133848, 18780741, 18780965, 37561482, 37561930, 121486365, 169028685, 242972730, 338057370, 360988056, 676114740, 1120584213, 1285201500, 1352229480, 2241168426, 2776831200, 5352575025, 5408917920, 7437262140, 10705150050

Offset: 1

Author

Benoit Cloitre, Apr 06 2002

Keywords

• nonn

Comments

Also k such that k divides A007429(k).

Also k such that k divides A211780(k). - Jaroslav Krizek, Sep 28 2014

a(28) > 10^10. - Giovanni Resta, Jun 10 2013

a(33) > 5*10^10. - Hiroaki Yamanouchi, Oct 05 2014

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..32
• Hiroaki Yamanouchi, Conjectured table of n, a(n) for n = 1..200

Crossrefs

Cf. A000005, A007429, A211780.

Programs

Mathematica

t = {}; n = 0; While[n++ <= 20000000, If[Mod[Total[DivisorSigma[1, Divisors[n]]], n] == 0, AppendTo[t, n]]]; t (* Jayanta Basu, Apr 03 2013 *)
f[p_, e_] := (p*(p^(e+1) - 1) - (p-1)*(e+1))/(p-1)^2; q[1] = True; q[k_] := Divisible[Times @@ f @@@ FactorInteger[k], k]; Select[Range[200000], q] (* Amiram Eldar, Apr 19 2025 *)

PARI

for(n=1, 20000000, if(denominator( sumdiv(n,d, numdiv(d)/d)) ==1, print1(n,",")))

PARI

isok(k) = {my(f = factor(k)); !(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p*(p^(e+1) - 1) - (p-1)*(e+1))/(p-1)^2) % k);} \\ Amiram Eldar, Apr 19 2025

Extensions

More terms from Rick L. Shepherd, Jun 23 2002

a(12)-a(27) from Giovanni Resta, Jun 10 2013

a(28) from Hiroaki Yamanouchi, Oct 05 2014