A380479 - OEIS

A380479

Number of prime factors (with repetition) in Product_{d|n} A276086(n/d)^A349394(d).

0, 1, 1, 3, 1, 3, 1, 8, 5, 4, 1, 7, 1, 3, 5, 18, 1, 7, 1, 11, 4, 5, 1, 14, 8, 4, 18, 9, 1, 8, 1, 40, 6, 6, 5, 17, 1, 5, 5, 24, 1, 9, 1, 15, 16, 7, 1, 32, 9, 13, 7, 13, 1, 21, 7, 22, 6, 8, 1, 15, 1, 3, 14, 82, 6, 9, 1, 15, 8, 10, 1, 30, 1, 4, 19, 13, 6, 10, 1, 52, 60, 6, 1, 21, 8, 5, 9, 32, 1, 18, 5, 19, 4, 7, 7, 62, 1, 15

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Note that A276150(.) [the sum of digits in the primorial base, A049345] in the second formula comes from A001222(A276086(.)).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to primorial base.

FORMULA

a(n) = A001222(A380459(n)).

a(n) = Sum_{p|n} Sum_{e=1..v_p(n)} (p^(e-1)) * A276150(n/(p^e)), where p ranges over the prime factors of n, and v_p(n) is the p-adic valuation of n, i.e., the maximal exponent h such that p^h | n.

PROG

(PARI)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A349394(n) = { my(p=0, e); if((e=isprimepower(n, &p)), p^(e-1), 0); };

A380459(n) = { my(m=1); fordiv(n, d, m *= A276086(d)^A349394(n/d)); (m); };

A380479(n) = bigomega(A380459(n));

(PARI)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };

A380479(n) = { my(f=factor(n)); sum(i=1, #f~, sum(e=1, f[i, 2], (f[i, 1]^(e-1))*A276150(n/(f[i, 1]^e)))); };

CROSSREFS

Cf. A001222, A049345, A276086, A276150, A349394, A380459.

Cf. also A380480.