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Permutation of natural numbers: If n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2*k, otherwise, when n is k-th number with an even number of prime divisors [i.e., n = A028260(k)], a(n) = (2*k)-1.
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%I #18 Mar 09 2019 07:23:17

%S 1,2,4,3,6,5,8,10,7,9,12,14,16,11,13,15,18,20,22,24,17,19,26,21,23,25,

%T 28,30,32,34,36,38,27,29,31,33,40,35,37,39,42,44,46,48,50,41,52,54,43,

%U 56,45,58,60,47,49,51,53,55,62,57,64,59,66,61,63,68,70,72,65,74,76,78

%N Permutation of natural numbers: If n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2*k, otherwise, when n is k-th number with an even number of prime divisors [i.e., n = A028260(k)], a(n) = (2*k)-1.

%C a(a(n)) = A143694(n).

%H Antti Karttunen, <a href="/A143692/b143692.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F From _Antti Karttunen_, Jul 27 2014: (Start)

%F If A066829(n) = 1, then a(n) = 2 * A055038(n), otherwise a(n) = (2 * A055037(n)) - 1.

%F For all n >= 1, A000035(a(n)) = 1 - A066829(n). [Permutation A245603 has the same property].

%F (End)

%p N:= 1000: # to get a(1) to a(N)

%p Odds,Evens:= selectremove(t -> numtheory:-bigomega(t)::odd,[$1..N]):

%p for k from 1 to nops(Odds) do A[Odds[k]]:= 2*k od:

%p for k from 1 to nops(Evens) do A[Evens[k]]:= 2*k-1 od:

%p seq(A[k],k=1..N); # _Robert Israel_, Jul 27 2014

%t m = 100;

%t odds = Select[Range[m], OddQ[PrimeOmega[#]]&];

%t evens = Select[Range[m], EvenQ[PrimeOmega[#]]&];

%t Do[a[odds[[k]]] = 2k, {k, 1, Length[odds]}];

%t Do[a[evens[[k]]] = 2k-1, {k, 1, Length[evens]}];

%t Array[a, m] (* _Jean-François Alcover_, Mar 09 2019, from Maple *)

%o (MIT/GNU Scheme) (define (A143692 n) (if (= 1 (A066829 n)) (* 2 (A055038 n)) (-1+ (* 2 (A055037 n)))))

%o (Haskell)

%o import Data.List (elemIndex); import Data.Maybe (fromJust)

%o a243692 = (+ 1) . fromJust . (`elemIndex` a143691_list)

%o -- _Reinhard Zumkeller_, Aug 07 2014

%Y Inverse: A143691.

%Y Cf. A000035, A055037, A055038, A066829, A143694, A245603.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Aug 29 2008

%E Name changed by _Antti Karttunen_, Jul 27 2014