A377986 - OEIS

%I #12 Nov 21 2024 15:30:21

%S 0,0,0,1,1,1,2,1,2,6,0,4,4,3,7

%N Number of integers k, with bigomega(k) > 2, whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

%C The solutions (composite, nonsemiprime antiderivatives of n!) are given in A377987.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n) = Sum_{k=1..A002620(n!)} [A003415(k) = n! and A001222(k) > 2], where [ ] is the Iverson bracket.

%F a(n) = A376410(n) - A062311(n).

%e See the examples in A377987.

%o (PARI)

%o A002620(n) = ((n^2)>>2);

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A377986(n) = { my(g=n!); sum(k=1,A002620(g),(bigomega(k)>2) && (A003415(k)==g)); };

%o (PARI) A377986(n) = AntiDeriv(n!,2,"a_terms_for_A377987_unsorted.txt"); \\ The rest of the program is given in A376410.

%Y Row lengths of irregular triangle A377987.

%Y Cf. A000142, A002620, A003415, A062311, A376410.

%K nonn,hard,more

%O 1,7

%A _Antti Karttunen_, Nov 19 2024