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A253564 Permutation of natural numbers: a(n) = A156552(A122111(n)).

%I A253564 #21 Jul 21 2017 16:01:46
%S A253564 0,1,3,2,7,5,15,4,6,11,31,9,63,23,13,8,127,10,255,19,27,47,511,17,14,
%T A253564 95,12,39,1023,21,2047,16,55,191,29,18,4095,383,111,35,8191,43,16383,
%U A253564 79,25,767,32767,33,30,22,223,159,65535,20,59,71,447
%N A253564 Permutation of natural numbers: a(n) = A156552(A122111(n)).
%C A253564 Note the indexing.
%C A253564 a(n) (n>=2) can be obtained by the composition of a bijection between {2,3,4,...} and the set of integer partitions and a bijection between the set of integer partitions and {1,2,3,4,...}. Explanation on the example n=18. Write 18 = 3*3*2 = 2'*2'*1', where m' = m-th prime. Consider the partition p = (2,2,1) and let b denote the southeast border of the Ferrers board of p. Form a binary number by replacing each east step of b by 1 and each north step of b, with the exception of the last one, by 0: 1010. Its value is a(18) = 10. - _Emeric Deutsch_, Sep 08 2016.
%H A253564 Alois P. Heinz, <a href="/A253564/b253564.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Antti Karttunen)
%H A253564 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A253564 a(n) = A156552(A122111(n)).
%F A253564 As a composition of other permutations:
%F A253564 a(n) = A054429(A253566(n)).
%p A253564 a:= proc(n) local i, l, r; r, l:= 0, [0, sort(map(i->
%p A253564       numtheory[pi](i[1])$i[2], ifactors(n)[2]))[]];
%p A253564       for i to nops(l)-1 do
%p A253564         r:= 2*((x-> 2*x+1)@@(l[i+1]-l[i]))(r)
%p A253564       od; r/2
%p A253564     end:
%p A253564 seq(a(n), n=1..120);  # _Alois P. Heinz_, Jul 21 2017
%t A253564 Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[ Table[ 2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &[If[n == 1, 1, Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ n]]], {n, 57}] (* _Michael De Vlieger_, Sep 08 2016, after _JungHwan Min_ at A122111 *)
%o A253564 (Scheme) (define (A253564 n) (A156552 (A122111 n)))
%Y A253564 Inverse: A253563.
%Y A253564 Cf. A054429, A122111, A156552, A161919, A253566.
%K A253564 nonn
%O A253564 1,3
%A A253564 _Antti Karttunen_, Jan 03 2015