

A243059


If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{kj} * p_{ki} * p_{kh} * ... * p_{ka} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.


6



1, 2, 3, 0, 5, 6, 7, 0, 0, 15, 11, 12, 13, 35, 10, 0, 17, 0, 19, 45, 21, 77, 23, 24, 0, 143, 0, 175, 29, 30, 31, 0, 55, 221, 14, 0, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 0, 0, 187, 1573, 53, 0, 33, 875, 247, 667, 59, 90, 61, 899, 63, 0, 65, 385, 67, 2873, 391, 70, 71, 0
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OFFSET

1,2


COMMENTS

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).
Differs from A243057 in that the "degenerate cases" A070003 are here zeros, but is otherwise equal to it (at the points given by A102750), i.e. for all n, a(A102750(n)) = A243057(A102750(n)) = A242420(A102750(n)).


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10001


FORMULA

a(1)=1, and for n>1, a(n) = q_{A243056(n)} * a(A032742(n)). Here q_{k} stands for 0 when k=0, and otherwise for the kth prime, A000040(k).
If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{kj} * p_{ki} * p_{kh} * ... * p_{ka} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k).


EXAMPLE

For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{33} * p_3 = 0*3 = 0. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 0 by the convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{31} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{21} * p{21} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{32} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{53} * p_{53} * p_{51} * p_{51} * p_{51} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{54} * p_{54} * p_{54} * p_{52} * p_{52} * p_5 = 2*2*2*5*5*11 = 2200.


PROG

(Scheme, with Antti Karttunen's IntSeqlibrary)
(definec (A243059 n) (if (<= n 1) n (* (if (zero? (A243056 n)) 0 (A000040 (A243056 n))) (A243059 (A032742 n)))))


CROSSREFS

Fixed points: A243058 (includes primes).
Positions of zeros: A070003.
Cf. A243057, A000040, A032742, A243056, A242419, A242420, A243286.
Sequence in context: A145105 A140700 A055615 * A190621 A325314 A322953
Adjacent sequences: A243056 A243057 A243058 * A243060 A243061 A243062


KEYWORD

nonn


AUTHOR

Antti Karttunen, May 31 2014


STATUS

approved



