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%I #24 Apr 19 2016 01:25:20
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,27,22,23,24,25,26,
%T 21,28,29,30,31,32,39,34,35,36,37,38,63,40,41,42,43,44,33,46,47,48,49,
%U 50,75,52,53,54,65,56,99,58,59,60,61,62,57,64,95,66,67,68,111,70,71,72,73,74,51,76,77,78,79,80,45,82,83,84,155,86,135
%N Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).
%C a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.
%C Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.
%H Antti Karttunen, <a href="/A249817/b249817.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(1) = 1, a(n) = A083221(A055396(n), A246277(n)).
%F a(1) = 1, a(n) = A083221(A055396(n), A249821(A055396(n), A078898(n))).
%F As a composition of other permutations:
%F a(1) = 1, and for n > 1, a(n) = 1 + A249815(n-1).
%F Other identities. For all n >= 1:
%F a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
%F A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].
%t lim = 87; a083221 = Table[Take[Prime[n] Select[Range[Ceiling[lim/2]^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], Ceiling[lim/2]], {n, Ceiling[lim/2]}]; a055396[n_] PrimePi[FactorInteger[n][[1, 1]]]; a246277[n_] := Which[n == 1, 0, EvenQ@ n, n/2, True, a246277[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ Transpose@ FactorInteger@ n, Last@ Transpose@ FactorInteger@ n]]]; Table[a083221[[a055396@ n, a246277@ n]], {n, 2, lim}] (* _Michael De Vlieger_, Jan 04 2016, after _Jean-François Alcover_ at A055396 and _Yasutoshi Kohmoto_ at A083140 *)
%o (Scheme) (define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A246277 n)))) ;; Code for A083221bi given in A083221
%o ;; Alternative version:
%o (define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A249821bi (A055396 n) (A078898 n))))) ;; Code for A249821bi given in A249821.
%Y Inverse: A249818.
%Y There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A246277(n), namely: A250245, A250247 and A250249.
%Y Other similar or related permutations: A249815.
%Y Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249821.
%Y Differs from its inverse A249818 for the first time at n=33, where a(33) = 39, while A249818(33) = 45.
%K nonn
%O 1,2
%A _Antti Karttunen_, Nov 06 2014
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