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

1,7

Comments

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

Links

Table of n, a(n) for n=1..15.

Index entries for sequences related to factorial numbers

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).

Example

See the examples in A377987.

Prog

(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.

Crossrefs

Row lengths of irregular triangle A377987.

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

Sequence in context: A000619 A006602 A144824 * A364513 A144358 A049404

Adjacent sequences: A377983 A377984 A377985 * A377987 A377988 A377989

Keyword

nonn,hard,more

Author

Antti Karttunen, Nov 19 2024

Status

approved