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

1,4

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.

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..., where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)). - Amiram Eldar, Oct 03 2023

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A368216 A335428 A368781 * A344417 A347437 A255231

Adjacent sequences: A050374 A050375 A050376 * A050378 A050379 A050380

Keyword

nonn,easy,mult

Author

Christian G. Bower, Nov 15 1999

Extensions

More terms from Antti Karttunen, Dec 28 2019

Status

approved