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A050377 Number of ways to factor n into "Fermi-Dirac primes" (members of A050376). 12
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 (list; graph; refs; listen; history; text; internal format)
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).

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

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000 (the first 10000 terms from Reinhard Zumkeller)

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.

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

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

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-Fran├žois 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, A018819, A050376-A050380, A025487.

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

Sequence in context: A046951 A159631 A335428 * A255231 A294874 A318324

Adjacent sequences:  A050374 A050375 A050376 * A050378 A050379 A050380

KEYWORD

nonn,mult

AUTHOR

Christian G. Bower, Nov 15 1999

EXTENSIONS

Data section extended up to a(105) by Antti Karttunen, Dec 28 2019

STATUS

approved

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Last modified April 21 17:36 EDT 2021. Contains 343156 sequences. (Running on oeis4.)