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%I #37 Nov 06 2021 11:07:19
%S 0,1,2,2,3,1,2,3,3,1,2,3,4,4,4,4,5,4,5,4,6,6,7,5,5,5,6,5,6,5,6,7,9,7,
%T 7,7,8,8,8,8,9,10,11,10,9,7,8,7,7,8,10,9,10,8,10,12,14,12,13,11,12,12,
%U 11,11,13,12,13,12,12,13,14,13,14,14,15,14,14,11,12,13,13,13,14,16,16,14
%N Number of primes with odd exponents in the prime power factorization of n!.
%C The products of the corresponding primes form A055204.
%C Also, the number of primes dividing the squarefree part of n! (=A055204(n)).
%C Also, the number of prime factors in the factorization of n! into distinct terms of A050376. See the references in A241289. - _Vladimir Shevelev_, Apr 16 2014
%D V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).
%H Max Alekseyev, <a href="/A055460/b055460.txt">Table of n, a(n) for n = 1..100000</a>
%H S. Litsyn and V. S. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%F a(n) = A001221(A055204(n)). - _Max Alekseyev_, Oct 19 2014
%F From _Wolfdieter Lang_, Nov 06 2021: (Start)
%F a(n) = A162642(A000142(n)).
%F a(n) = A000720(n) - A348841(n), (End)
%e For n = 100, the exponents of primes in the factorization of n! are {97,48,24,16,9,7,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1}, and there are 17 odd values: {97,9,7,5,5,3,3,1,1,1,1,1,1,1,1,1,1}, so a(100) = 17.
%e The factorization of 6! into distinct terms of A050376 is 5*9*16 with only one prime, so a(6)=1. - _Vladimir Shevelev_, Apr 16 2014
%t Table[Count[FactorInteger[n!][[All, -1]], m_ /; OddQ@ m] - Boole[n == 1], {n, 100}] (* _Michael De Vlieger_, Feb 05 2017 *)
%o (PARI) a(n) = omega(core(n!))
%Y Cf. A000142, A000720, A007913, A008833, A162642, A348841.
%Y Cf. A249016 (indices of records), A249017 (values of records)
%K nonn
%O 1,3
%A _Labos Elemer_, Jun 26 2000
%E Edited by _Max Alekseyev_, Oct 19 2014
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