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A375635 The number of infinitary divisors of n!.

%I A375635 #17 Aug 22 2024 16:46:51
%S A375635 1,1,2,4,8,16,8,16,64,64,16,32,128,256,512,1024,2048,4096,256,512,512,
%T A375635 2048,4096,8192,8192,16384,32768,65536,16384,32768,65536,131072,
%U A375635 524288,2097152,131072,65536,32768,65536,131072,262144,524288,1048576,4194304,8388608
%N A375635 The number of infinitary divisors of n!.
%H A375635 Amiram Eldar, <a href="/A375635/b375635.txt">Table of n, a(n) for n = 0..3000</a>
%F A375635 a(n) = A037445(n!).
%F A375635 a(n) = 2^A177329(n).
%F A375635 A048656(n) <= a(n) <= A027423(n).
%t A375635 f[p_, e_] := 2^DigitCount[e, 2, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 100, 0]
%o A375635 (PARI) a(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n!)[,2]));
%o A375635 (Python)
%o A375635 from collections import Counter
%o A375635 from sympy import factorint
%o A375635 def A375635(n): return 1<<sum(e.bit_count() for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # _Chai Wah Wu_, Aug 22 2024
%Y A375635 Cf. A027423, A037445, A048656, A177329.
%K A375635 nonn
%O A375635 0,3
%A A375635 _Amiram Eldar_, Aug 22 2024