%I #17 Apr 28 2022 13:33:15

%S 1,2,2,2,2,3,2,2,2,3,2,4,2,3,3,2,2,4,2,4,3,3,2,4,2,3,2,4,2,5,2,2,3,3,

%T 3,3,2,3,3,4,2,5,2,4,4,3,2,4,2,4,3,4,2,4,3,4,3,3,2,6,2,3,4,2,3,5,2,4,

%U 3,5,2,4,2,3,4,4,3,5,2,4,2,3,2,6,3,3,3,4,2,6,3,4,3,3,3,4,2,4,4,3,2,5,2,4,5

%N The second prime shadow of n: a(1) = 1; for n > 1, a(n) = a(A003557(n)) * prime(A056169(n)) when A056169(n) > 0, otherwise a(n) = a(A003557(n)).

%C a(n) depends only on prime signature of n (cf. A025487).

%H Antti Karttunen, <a href="/A328830/b328830.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A328830/a328830.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(1) = 1; for n > 1, a(n) = A008578(1+A056169(n)) * a(A003557(n)).

%F A001221(a(n)) = A323022(n).

%F A001222(a(n)) = A071625(n).

%F a(n) = A181819(A181819(n)). - _Gus Wiseman_, Apr 27 2022

%e For n = 30 = 2 * 3 * 5, there are three unitary prime factors, while A003557(30) = 1, which terminates the recursion, thus a(30) = prime(3) = 5.

%e For n = 60060 = 2^2 * 3 * 5 * 7 * 11 * 13, there are 5 unitary prime factors, while in A003557(60060) = 2 there are one, thus a(60060) = prime(5) * prime(1) = 11 * 2 = 22.

%e The number 1260 = 2^2*3^2*5*7 has prime exponents (2,2,1,1) so its prime shadow is prime(2)*prime(2)*prime(1)*prime(1) = 36. Next, 36 = 2^2*3^2 has prime exponents (2,2) so its prime shadow is prime(2)*prime(2) = 9. In fact, the term a(1260) = 9 is the first appearance of 9 in the sequence. - _Gus Wiseman_, Apr 28 2022

%o (PARI)

%o A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557

%o A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ From A056169

%o A328830(n) = if(1==n,n, my(u=A056169(n)); if(0==u,1,prime(u)) * A328830(A003557(n)));

%Y Column 2 of A353510.

%Y Cf. A001221, A001222, A003557, A008578, A056169, A071625, A323022.

%Y Differs from A182860 for the first time at a(30) = 5, while A182860(30) = 4.

%Y Cf. A182863 for the first appearances.

%Y A005361 gives product of prime exponents.

%Y A112798 gives prime indices, sum A056239.

%Y A124010 gives prime signature, sorted A118914.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A325131 lists numbers relatively prime to their prime shadow.

%Y A325755 lists numbers divisible by their prime shadow.

%Y Cf. A109297, A182850, A353393, A353507.

%K nonn

%O 1,2

%A _Antti Karttunen_, Oct 29 2019

%E Added _Gus Wiseman_'s new name to the front of the definition. - _Antti Karttunen_, Apr 27 2022