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%I #21 May 09 2017 22:40:40
%S 1,2,4,3,9,8,6,5,25,18,16,27,15,12,10,7,49,50,36,75,81,32,54,125,35,
%T 30,24,45,21,20,14,11,121,98,100,147,225,72,150,245,625,162,64,243,
%U 375,108,250,343,77,70,60,105,135,48,90,175,55,42,40,63,33,28,22,13,169,242,196,363,441,200,294,605,1225,450,144
%N Permutation of natural numbers which maps between the partitions as encoded in A227739 (binary based system, zero-based) to A112798 (prime-index based system, one-based).
%C Note the indexing: the domain includes zero, but the range starts from one.
%H Antti Karttunen, <a href="/A243353/b243353.txt">Table of n, a(n) for n = 0..8192</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = A005940(1+A003188(n)).
%F a(n) = A241909(1+A075157(n)). [With A075157's original starting offset]
%F For all n >= 0, A243354(a(n)) = n.
%F A227183(n) = A056239(a(n)). [Maps between the corresponding sums ...]
%F A227184(n) = A003963(a(n)). [... and products of parts of each partition].
%F For n >= 0, a(A037481(n)) = A002110(n). [Also "triangular partitions", the fixed points of Bulgarian solitaire, A226062 & A242424].
%F For n >= 1, a(A227451(n+1)) = 4*A243054(n).
%t f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Table[f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, 74}] (* _Michael De Vlieger_, May 09 2017 *)
%o (Scheme) (define (A243353 n) (A005940 (+ 1 (A003188 n))))
%o (Python)
%o from sympy import prime
%o import math
%o def A(n): return n - 2**int(math.floor(math.log(n, 2)))
%o def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
%o def a005940(n): return b(n - 1)
%o def a003188(n): return n^int(n/2)
%o def a243353(n): return a005940(1 + a003188(n)) # _Indranil Ghosh_, May 07 2017
%Y A243354 gives the inverse mapping.
%Y Cf. A227739, A112798, A075157, A241909, A005940, A003188, A226062, A242424.
%K nonn
%O 0,2
%A _Antti Karttunen_, Jun 05 2014
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