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%I #51 Oct 03 2023 13:11:35
%S 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,
%T 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,
%U 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
%N Number of ways to factor n into "Fermi-Dirac primes" (members of A050376).
%C 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).
%H Antti Karttunen, <a href="/A050377/b050377.txt">Table of n, a(n) for n = 1..100000</a> (the first 10000 terms from Reinhard Zumkeller)
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F Dirichlet g.f.: Product_{n in A050376} (1/(1-1/n^s)).
%F a(p^k) = A000123([k/2]) for all primes p.
%F a(A002110(n)) = 1.
%F Multiplicative with a(p^e) = A018819(e). - _Christian G. Bower_ and _David W. Wilson_, May 22 2005
%F a(n) = Sum{a(d): d^2 divides n}, a(1) = 1. - _Reinhard Zumkeller_, Jul 12 2007
%F a(A108951(n)) = A330690(n). - _Antti Karttunen_, Dec 28 2019
%F a(n) = 1 for all squarefree values of n (A005117). - _Eric Fox_, Feb 03 2020
%F G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^(k^2)). - _Ilya Gutkovskiy_, Nov 25 2020
%F 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
%p A018819:= proc(n) option remember;
%p if n::odd then procname(n-1)
%p else procname(n-1) + procname(n/2)
%p fi
%p end proc:
%p A018819(0):= 1:
%p f:= n -> mul(A018819(s[2]),s=ifactors(n)[2]):
%p seq(f(n),n=1..100); # _Robert Israel_, Jan 14 2016
%t b[0] = 1; b[n_] := b[n] = b[n - 1] + If[EvenQ[n], b[n/2], 0];
%t a[n_] := Times @@ (b /@ FactorInteger[n][[All, 2]]);
%t Array[a, 102] (* _Jean-François Alcover_, Jan 27 2018 *)
%o (PARI)
%o A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
%o A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2])); \\ _Antti Karttunen_, Dec 28 2019
%Y Cf. A000123, A001055, A002110, A005117, A018819, A050376-A050380, A025487.
%Y Cf. A108951, A330687 (positions of records), A330688 (record values), A330689, A330690.
%K nonn,easy,mult
%O 1,4
%A _Christian G. Bower_, Nov 15 1999
%E More terms from _Antti Karttunen_, Dec 28 2019
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