This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348841 #23 Nov 28 2021 02:55:05
%S A348841 0,0,0,0,0,2,2,1,1,3,3,2,2,2,2,2,2,3,3,4,2,2,2,4,4,4,3,4,4,5,5,4,2,4,
%T A348841 4,4,4,4,4,4,4,3,3,4,5,7,7,8,8,7,5,6,6,8,6,4,2,4,4,6,6,6,7,7,5,6,6,7,
%U A348841 7,6,6,7,7,7,6,7,7,10,10,9
%N A348841 Number of primes with even exponents >= 2 in the prime power factorization of n!, for n >= 1.
%C A348841 The restriction to positive exponents in the prime factor factorization is used to avoid the ambiguity due to p^0 = 1 for any prime. Then a(n) = A000720(n) - A055460(n), for n >= 1.
%H A348841 Amiram Eldar, <a href="/A348841/b348841.txt">Table of n, a(n) for n = 1..10000</a>
%F A348841 a(n) = A000720(n) - A055460(n), for n >= 1.
%F A348841 a(n) = A162641(A000142(n)). - _Michel Marcus_, Nov 03 2021
%e A348841 n = 12: 12! = 479001600 = 2^10 * 5^2 * 3^5 * 7^1 * 11^1, hence a(12) = 2, A055460(12) = 3 and A000720(12) = 5. This latter equation holds because 2, 3, 5, 7, 11 are the primes not exceeding 12.
%t A348841 Table[Length@Select[FactorInteger[n!],EvenQ@Last@#&],{n,80}] (* _Giorgos Kalogeropoulos_, Nov 02 2021 *)
%o A348841 (PARI) a(n) = my(f=factor(n!)); #select(x->(! (x%2)), f[,2]); \\ _Michel Marcus_, Nov 03 2021
%o A348841 (PARI) a(n) = my(res = 0); forprime(p = 2, n\2, res+=(val(n, p)%2==0)); res
%o A348841 val(n, p) = my(r=0); while(n, r+=n\=p); r \\ _David A. Corneth_, Nov 03 2021
%Y A348841 Cf. A000720, A055460.
%Y A348841 Cf. A000142, A162641.
%K A348841 nonn
%O A348841 1,6
%A A348841 _Wolfdieter Lang_, Nov 02 2021