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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). 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 25 15:08 EST 2021. Contains 340416 sequences. (Running on oeis4.)