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%I #14 May 08 2022 10:34:31
%S 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,
%T 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,
%U 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
%N Prime shadow of primorial base exp-function: a(n) = A181819(A276086(n)).
%C From _Antti Karttunen_, Apr 30 2022: (Start)
%C 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.
%C 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.
%C (End)
%H Antti Karttunen, <a href="/A328835/b328835.txt">Table of n, a(n) for n = 0..30030</a>
%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F a(n) = A181819(A276086(n)).
%F A001222(a(n)) = A267263(n).
%F A007814(a(n)) = A328614(n).
%F A061395(a(n)) = A328114(n).
%F For all n >= 0, a(n) = A181819(A278226(n)) and A181821(a(n)) = A278226(n). - _Antti Karttunen_, Apr 30 2022
%o (PARI)
%o A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A328835(n) = A181819(A276086(n));
%Y Cf. A001222, A007814, A061395, A181819, A181821, A267263, A276086, A278226 (A286626), A328114, A328614.
%Y Cf. A275735 for analogous sequence.
%K nonn
%O 0,2
%A _Antti Karttunen_, Oct 29 2019
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