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A195349 Numbers n such that Sum_{k=1..n} d(k) divides Product_{k=1..n} d(k), where d(k) is the number of divisors of k.

%I A195349 #21 Oct 10 2021 04:13:11
%S A195349 1,7,19,41,57,64,68,133,145,149,164,235,267,291,317,336,358,419,433,
%T A195349 503,528,566,599,612,659,726,801,927,1017,1035,1077,1118,1190,1206,
%U A195349 1213,1281,1297,1309,1320,1323,1367,1446,1473,1485,1516,1595,1611,1634,1941
%N A195349 Numbers n such that Sum_{k=1..n} d(k) divides Product_{k=1..n} d(k), where d(k) is the number of divisors of k.
%C A195349 d(k) is sometimes called tau(k) or sigma_0(k). Is this sequence infinite?
%H A195349 Chai Wah Wu, <a href="/A195349/b195349.txt">Table of n, a(n) for n = 1..2000</a> (term 1..500 from Harvey P. Dale)
%t A195349 t = {}; a = 0; b = 1; Do[a = a + DivisorSigma[0, n]; b = b*DivisorSigma[0, n]; If[Mod[b, a] == 0, AppendTo[t, n]], {n, 2000}]; t (* _T. D. Noe_, Sep 16 2011 *)
%t A195349 With[{c=DivisorSigma[0,Range[2000]]},Position[Thread[{FoldList[ Times,c], Accumulate[ c]}],_?(Divisible[#[[1]],#[[2]]]&),1,Heads->False]] // Flatten (* _Harvey P. Dale_, Apr 14 2019 *)
%o A195349 (Python)
%o A195349 from sympy import divisor_count
%o A195349 A195349_list, s, p = [], 0, 1
%o A195349 for k in range(1,10**4):
%o A195349     d = divisor_count(k)
%o A195349     s += d
%o A195349     p *= d
%o A195349     if p % s == 0:
%o A195349         A195349_list.append(k) # _Chai Wah Wu_, Oct 09 2021
%Y A195349 Cf. A000005, A006218, A066843.
%K A195349 nonn
%O A195349 1,2
%A A195349 _Carl Najafi_, Sep 16 2011