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%I #28 Jun 02 2014 22:24:32
%S 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,
%T 175,29,30,31,2,55,221,14,12,37,323,91,135,41,105,43,539,20,437,47,48,
%U 7,15,187,1573,53,6,33,875,247,667,59,90,61,899,63,2,65,385
%N 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_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).
%C A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).
%C A102750 gives such n that a(a(n)) = n. A243286 is the self-inverse permutation induced when the domain is restricted to A102750. Cf. also A242420.
%C 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.
%H Antti Karttunen, <a href="/A243057/b243057.txt">Table of n, a(n) for n = 1..10001</a>
%F 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_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).
%F 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 k-th prime, A000040(k).
%F For all n, a(n) = a(A243074(n)).
%F For all k in A102750, a(k) = A242420(k).
%e For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * 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.]
%e For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
%e For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
%e For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
%e 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_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
%e 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_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.
%o (Scheme with _Antti Karttunen_'s IntSeq-library)
%o (definec (A243057 n) (if (<= n 1) n (* (if (zero? (A243056 n)) 1 (A000040 (A243056 n))) (A243057 (A032742 n)))))
%Y Fixed points: A243058 (includes primes).
%Y Cf. A243059, A000040, A032742, A243056, A242419, A242420, A243286, A070003, A102750, A243074.
%K nonn
%O 1,2
%A _Antti Karttunen_, May 31 2014
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