A328835 Prime shadow of primorial base exp-function: a(n) = A181819(A276086(n)).

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 5, 10, 10, 20, 15, 30, 7, 14, 14, 28, 21, 42, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 10, 20, 20, 40, 30, 60, 14, 28, 28, 56, 42, 84, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 15, 30, 30, 60, 45, 90, 21, 42, 42, 84, 63, 126, 5, 10, 10, 20, 15, 30, 10, 20, 20

Offset: 0

Author

Antti Karttunen, Oct 29 2019

Keywords

• nonn

Comments

From Antti Karttunen, Apr 30 2022: (Start)

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of times a nonzero digit k occurs in the primorial base representation of n.

Note that this sequence, and all the sequences derived from it as b(n) = f(a(n)), [where f is any integer-valued function] can be represented as b(n) = g(A278226(n)), where g(n) = f(A181819(n)). E.g., if f is the identity function (so that b(n) is this sequence), then g(n) is A181819(n). See the comment and formulas in the latter sequence.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 0..30030
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A001222, A007814, A061395, A181819, A181821, A267263, A276086, A278226 (A286626), A328114, A328614.

Cf. A275735 for analogous sequence.

Programs

PARI

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328835(n) = A181819(A276086(n));

Formula

a(n) = A181819(A276086(n)).

A001222(a(n)) = A267263(n).

A007814(a(n)) = A328614(n).

A061395(a(n)) = A328114(n).

For all n >= 0, a(n) = A181819(A278226(n)) and A181821(a(n)) = A278226(n). - Antti Karttunen, Apr 30 2022