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A328051 Numbers m such that sigma(m)/(d(m)*sopf(m)) is an integer, where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

%I A328051 #15 Sep 08 2022 08:46:24
%S A328051 20,35,42,54,140,189,195,207,209,276,378,464,470,500,506,510,527,540,
%T A328051 608,660,672,741,846,864,875,899,923,945,989,1029,1120,1276,1316,1323,
%U A328051 1334,1349,1365,1519,1539,1564,1595,1715,1725,1736,1755,1815,1880,1887,1914,2058
%N A328051 Numbers m such that sigma(m)/(d(m)*sopf(m)) is an integer, where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).
%C A328051 This sequence is motivated by the short fate of A134382.
%H A328051 Michel Marcus, <a href="/A328051/b328051.txt">Table of n, a(n) for n = 1..10000</a>
%e A328051 For n=20, sigma(20)/(d(20)*sopf(20)) = 42/(6*7) = 1, an integer, so 20 is a term.
%t A328051 f[p_, e_] := (p^(e + 1) - 1)/((e + 1)*(p - 1)); Select[Range[2, 2100], IntegerQ[ Times @@ (f @@@ (fct = FactorInteger[#])) / Plus @@ (fct[[;; , 1]])] &] (* _Amiram Eldar_, Oct 03 2019 *)
%o A328051 (PARI) sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
%o A328051 isok(m) = if (m>1, my(f=factor(m)); (sigma(f) % (numdiv(f)*sopf(f))) == 0);
%o A328051 (Magma) [k: k in [2..2100]|IsIntegral(DivisorSigma(1,k)/(#Divisors(k)*(&+PrimeDivisors(k))))]; // _Marius A. Burtea_, Oct 03 2019
%Y A328051 Cf. A000005, A000203, A008472, A134382, A328052.
%K A328051 nonn
%O A328051 1,1
%A A328051 _Michel Marcus_, Oct 03 2019