A050226 Numbers m such that m divides Sum_{k = 1..m} A000005(k).

1, 4, 5, 15, 42, 44, 47, 121, 336, 340, 347, 930, 2548, 6937, 6947, 51322, 379097, 379131, 379133, 2801205, 20698345, 56264090, 56264197, 152941920, 152942012, 8350344420, 61701166395, 455913379395, 455913379831, 1239301050694, 3368769533660, 3368769533812

Offset: 1

Author

Labos Elemer, Dec 20 1999

Keywords

• nonn
• nice

Examples

			For k = 15 the sum is 1 + 2 + 2 + 3 + 2 + 4 + 2 + 4 + 3 + 4 + 2 + 6 + 2 + 4 + 4 = 45 which is divisible by 15.
		

References

• Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..39 (quotients <= 40)

Crossrefs

Cf. A000005, A006218, A057494, A085567, A085829.

Programs

Mathematica

s = 0; Do[ s = s + DivisorSigma[ 0, n ]; If[ Mod[ s, n ] == 0, Print[ n ] ], {n, 1, 2*10^9} ]
k=10^6; a[1]=1;a[n_]:=a[n]=DivisorSigma[0,n]+a[n-1]; nd=a/@Range@k; Select[Range@k,Divisible[nd[[#]],#]&] (* Ivan N. Ianakiev, Apr 30 2016 *)
Module[{nn=400000},Select[Thread[{Range[nn],Accumulate[DivisorSigma[0,Range[nn]]]}],Divisible[#[[2]],#[[1]]]&]][[All,1]] (* The program generates the first 19 terms of the sequence. To generate more, increase the nn constant. *) (* Harvey P. Dale, Jul 03 2022 *)

PARI

lista(nn) = {my(s = 0); for (n=1, nn, s += numdiv(n); if (!(s % n), print1(n, ", ")););} \\ Michel Marcus, Dec 14 2015

Sage

def A050226_list(len):
    a, L = 0, []
    for n in (1..len):
        a += sigma(n,0)
        if n.divides(a): L.append(n)
    return L
A050226_list(10000) # Peter Luschny, Dec 18 2015

Formula

m is in the sequence if Sum_{i = 1..m} d(i) = m*k, k an integer, where d(i) = number of divisors of i.

Extensions

More terms from Robert G. Wilson v, Sep 21 2000

Further terms from Naohiro Nomoto, Aug 03 2001

a(26)-a(30) from Donovan Johnson, Dec 21 2008