A377986 Number of integers k, with bigomega(k) > 2, whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

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

Offset: 1

Author

Antti Karttunen, Nov 19 2024

Keywords

• nonn
• hard
• more

Comments

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

Examples

			See the examples in A377987.
		

Links

• Index entries for sequences related to factorial numbers

Crossrefs

Row lengths of irregular triangle A377987.

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

Programs

PARI

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A377986(n) = { my(g=n!); sum(k=1,A002620(g),(bigomega(k)>2) && (A003415(k)==g)); };

PARI

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

Formula

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

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