A328830 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)).

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, 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, 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

Offset

1,2

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

Formula

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

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

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

a(n) = A181819(A181819(n)). - Gus Wiseman, Apr 27 2022

Example

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.
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.
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

Prog

(PARI)
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ From A056169
A328830(n) = if(1==n, n, my(u=A056169(n)); if(0==u, 1, prime(u)) * A328830(A003557(n)));

Crossrefs

Column 2 of A353510.

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

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

Cf. A182863 for the first appearances.

A005361 gives product of prime exponents.

A112798 gives prime indices, sum A056239.

A124010 gives prime signature, sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A325131 lists numbers relatively prime to their prime shadow.

A325755 lists numbers divisible by their prime shadow.

Cf. A109297, A182850, A353393, A353507.

Sequence in context: A062843 A136164 A182860 * A033947 A069719 A316990

Adjacent sequences: A328827 A328828 A328829 * A328831 A328832 A328833

Keyword

nonn

Author

Antti Karttunen, Oct 29 2019

Extensions

Added Gus Wiseman's new name to the front of the definition. - Antti Karttunen, Apr 27 2022

Status

approved