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).

20, 35, 42, 54, 140, 189, 195, 207, 209, 276, 378, 464, 470, 500, 506, 510, 527, 540, 608, 660, 672, 741, 846, 864, 875, 899, 923, 945, 989, 1029, 1120, 1276, 1316, 1323, 1334, 1349, 1365, 1519, 1539, 1564, 1595, 1715, 1725, 1736, 1755, 1815, 1880, 1887, 1914, 2058

Offset

1,1

Comments

This sequence is motivated by the short fate of A134382.

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Example

For n=20, sigma(20)/(d(20)*sopf(20)) = 42/(6*7) = 1, an integer, so 20 is a term.

Mathematica

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 *)

Prog

(PARI) sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
isok(m) = if (m>1, my(f=factor(m)); (sigma(f) % (numdiv(f)*sopf(f))) == 0);
(Magma) [k: k in [2..2100]|IsIntegral(DivisorSigma(1, k)/(#Divisors(k)*(&+PrimeDivisors(k))))]; // Marius A. Burtea, Oct 03 2019

Crossrefs

Cf. A000005, A000203, A008472, A134382, A328052.

Sequence in context: A157426 A325603 A024747 * A081962 A024755 A048022

Adjacent sequences: A328048 A328049 A328050 * A328052 A328053 A328054

Keyword

nonn

Author

Michel Marcus, Oct 03 2019

Status

approved