A050377 Number of ways to factor n into "Fermi-Dirac primes" (members of A050376).

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 4, 1, 1, 1, 2, 1

Offset: 1

Author

Christian G. Bower, Nov 15 1999

Keywords

• nonn
• easy
• mult

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000 (the first 10000 terms from Reinhard Zumkeller)
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000123, A001055, A002110, A005117, A018819, A050376-A050380, A025487.

Cf. A108951, A330687 (positions of records), A330688 (record values), A330689, A330690, A382295.

Programs

Maple

A018819:= proc(n) option remember;
  if n::odd then procname(n-1)
  else procname(n-1) + procname(n/2)
  fi
end proc:
A018819(0):= 1:
f:= n -> mul(A018819(s[2]),s=ifactors(n)[2]):
seq(f(n),n=1..100); # Robert Israel, Jan 14 2016

Mathematica

b[0] = 1; b[n_] := b[n] = b[n - 1] + If[EvenQ[n], b[n/2], 0];
a[n_] := Times @@ (b /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 27 2018 *)

PARI

A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2])); \\ Antti Karttunen, Dec 28 2019

Formula

Dirichlet g.f.: Product_{n in A050376} (1/(1-1/n^s)).

a(p^k) = A000123([k/2]) for all primes p.

a(A002110(n)) = 1.

Multiplicative with a(p^e) = A018819(e). - Christian G. Bower and David W. Wilson, May 22 2005

a(n) = Sum{a(d): d^2 divides n}, a(1) = 1. - Reinhard Zumkeller, Jul 12 2007

a(A108951(n)) = A330690(n). - Antti Karttunen, Dec 28 2019

a(n) = 1 for all squarefree values of n (A005117). - Eric Fox, Feb 03 2020

G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Nov 25 2020

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.7876368001694456669... (A382295), where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)). - Amiram Eldar, Oct 03 2023

Extensions

More terms from Antti Karttunen, Dec 28 2019