cp's OEIS Frontend

A243353 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).

%I A243353 #21 May 09 2017 22:40:40
%S A243353 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 A243353 30,24,45,21,20,14,11,121,98,100,147,225,72,150,245,625,162,64,243,
%U A243353 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 A243353 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 A243353 Note the indexing: the domain includes zero, but the range starts from one.
%H A243353 Antti Karttunen, <a href="/A243353/b243353.txt">Table of n, a(n) for n = 0..8192</a>
%H A243353 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A243353 a(n) = A005940(1+A003188(n)).
%F A243353 a(n) = A241909(1+A075157(n)). [With A075157's original starting offset]
%F A243353 For all n >= 0, A243354(a(n)) = n.
%F A243353 A227183(n) = A056239(a(n)). [Maps between the corresponding sums ...]
%F A243353 A227184(n) = A003963(a(n)). [... and products of parts of each partition].
%F A243353 For n >= 0, a(A037481(n)) = A002110(n). [Also "triangular partitions", the fixed points of Bulgarian solitaire, A226062 & A242424].
%F A243353 For n >= 1, a(A227451(n+1)) = 4*A243054(n).
%t A243353 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 A243353 (Scheme) (define (A243353 n) (A005940 (+ 1 (A003188 n))))
%o A243353 (Python)
%o A243353 from sympy import prime
%o A243353 import math
%o A243353 def A(n): return n - 2**int(math.floor(math.log(n, 2)))
%o A243353 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 A243353 def a005940(n): return b(n - 1)
%o A243353 def a003188(n): return n^int(n/2)
%o A243353 def a243353(n): return a005940(1 + a003188(n)) # _Indranil Ghosh_, May 07 2017
%Y A243353 A243354 gives the inverse mapping.
%Y A243353 Cf. A227739, A112798, A075157, A241909, A005940, A003188, A226062, A242424.
%K A243353 nonn
%O A243353 0,2
%A A243353 _Antti Karttunen_, Jun 05 2014