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

%I A050226 #41 Feb 19 2024 01:56:50
%S A050226 1,4,5,15,42,44,47,121,336,340,347,930,2548,6937,6947,51322,379097,
%T A050226 379131,379133,2801205,20698345,56264090,56264197,152941920,152942012,
%U A050226 8350344420,61701166395,455913379395,455913379831,1239301050694,3368769533660,3368769533812
%N A050226 Numbers m such that m divides Sum_{k = 1..m} A000005(k).
%D A050226 Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.
%H A050226 Donovan Johnson, <a href="/A050226/b050226.txt">Table of n, a(n) for n = 1..39</a> (quotients <= 40)
%F A050226 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.
%e A050226 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.
%t A050226 s = 0; Do[ s = s + DivisorSigma[ 0, n ]; If[ Mod[ s, n ] == 0, Print[ n ] ], {n, 1, 2*10^9} ]
%t A050226 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 *)
%t A050226 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 *)
%o A050226 (PARI) lista(nn) = {my(s = 0); for (n=1, nn, s += numdiv(n); if (!(s % n), print1(n, ", ")););} \\ _Michel Marcus_, Dec 14 2015
%o A050226 (Sage)
%o A050226 def A050226_list(len):
%o A050226     a, L = 0, []
%o A050226     for n in (1..len):
%o A050226         a += sigma(n,0)
%o A050226         if n.divides(a): L.append(n)
%o A050226     return L
%o A050226 A050226_list(10000) # _Peter Luschny_, Dec 18 2015
%Y A050226 Cf. A000005, A006218, A057494, A085567, A085829.
%K A050226 nonn,nice
%O A050226 1,2
%A A050226 _Labos Elemer_, Dec 20 1999
%E A050226 More terms from _Robert G. Wilson v_, Sep 21 2000
%E A050226 Further terms from _Naohiro Nomoto_, Aug 03 2001
%E A050226 a(26)-a(30) from _Donovan Johnson_, Dec 21 2008