

A243057


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} = 1 and for k>=1, p_{k} = A000040(k).


8



1, 2, 3, 2, 5, 6, 7, 2, 3, 15, 11, 12, 13, 35, 10, 2, 17, 6, 19, 45, 21, 77, 23, 24, 5, 143, 3, 175, 29, 30, 31, 2, 55, 221, 14, 12, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 7, 15, 187, 1573, 53, 6, 33, 875, 247, 667, 59, 90, 61, 899, 63, 2, 65, 385
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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).
A102750 gives such n that a(a(n)) = n. A243286 is the selfinverse permutation induced when the domain is restricted to A102750. Cf. also A242420.
A070003 gives all n such that a(a(n)) <> n. Another variant, A243059, is defined to be zero when n is one of the terms of A070003.


LINKS

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


FORMULA

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} = 1 and for k>=1, p_{k} = A000040(k).
a(1)=1, and for n>1, a(n) = p_{A243056(n)} * a(A032742(n)). Here p_{k} stands for 1 when k=0, and otherwise for the kth prime, A000040(k).
For all n, a(n) = a(A243074(n)).
For all k in A102750, a(k) = A242420(k).


EXAMPLE

For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{33} * p_3 = 1*3 = 3. [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 1 by 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 (A243057 n) (if (<= n 1) n (* (if (zero? (A243056 n)) 1 (A000040 (A243056 n))) (A243057 (A032742 n)))))


CROSSREFS

Fixed points: A243058 (includes primes).
Cf. A243059, A000040, A032742, A243056, A242419, A242420, A243286, A070003, A102750, A243074.
Sequence in context: A175781 A072775 A304768 * A090078 A325814 A325126
Adjacent sequences: A243054 A243055 A243056 * A243058 A243059 A243060


KEYWORD

nonn


AUTHOR

Antti Karttunen, May 31 2014


STATUS

approved



