A392608 Number of nonzero digits in the primorial base representation of n (A049345) such that their corresponding radix prime divides n.

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 3

Offset

0,16

Comments

For n > 0, a(n) is the number of prime factors p of n such that the primorial base representation of n (A049345) has a nonzero digit under that same radix p, which is equal to the number of prime factors of n that divide A276086(n) [or vice versa], i.e., number of prime factors shared by n and A276806(n).

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to primorial base.

Formula

a(0) = 0, for n > 0, a(n) = Sum_{p|n, p prime} [A276086(n) == 0 (mod p)], where [ ] is the Iverson bracket.

a(n) = A001221(A324198(n)).

a(n) = A267263(n) - A392610(n).

For n > 0, a(n) = A001221(n) - A392609(n).

Example

a(0) = 0 as A049345(0) = 0 does not contain any nonzero digits.
a(1) = 0 as A049345(1) = 1, so the corresponding radix prime is 2, and 2 does not divide 1.
a(15) = 2 as A049345(15) = 211, with the corresponding radix primes 2, 3 and 5, of which 3 and 5 divide 6.
a(140) = 2 as A049345(140) = 4310, with nonzero digits having radix primes 3, 5, 7, of which the two latter divide 140 = 2^2 * 5 * 7.
a(392) = 1 as A049345(392) = 16010 [thus A276086(392) = 11^1 * 7^6 * 3^1], with nonzero digits having radix primes 11, 7, 3, of which only 7 divides 392 = 2^3 * 7^2.

Prog

(PARI) A392608(n) = { my(p=2, s=0, orgn=n); while(n, s += ((n%p) && !(orgn%p)); n = n\p; p = nextprime(1+p)); (s); };

Crossrefs

Cf. A001221, A049345, A267263, A276086, A324198, A392609, A392610.

Cf. A324583 (positions of 0's), A324584 (of terms > 0).

Sequence in context: A128616 A331902 A333817 * A270417 A353333 A353303

Adjacent sequences: A392605 A392606 A392607 * A392609 A392610 A392611

Keyword

nonn,base,easy

Author

Antti Karttunen, Jan 24 2026

Status

approved