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A254118 Permutation of natural numbers: a(n) = A249745(1+A254103(n)) - 1.

%I A254118 #22 May 06 2021 16:31:47
%S A254118 1,2,3,6,5,4,8,20,11,7,9,33,18,23,14,13,30,36,21,44,10,29,15,55,53,28,
%T A254118 16,74,39,41,12,179,90,96,50,114,24,42,35,92,69,47,19,86,25,51,26,236,
%U A254118 153,110,81,101,22,45,48,221,113,119,56,77,65,38,17,546,182
%N A254118 Permutation of natural numbers: a(n) = A249745(1+A254103(n)) - 1.
%H A254118 Antti Karttunen, <a href="/A254118/b254118.txt">Table of n, a(n) for n = 1..8191</a>
%H A254118 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A254118 a(n) = A249745(1+A254103(n)) - 1.
%F A254118 a(n) = (A254116((2*n)+1)-1) / 2. [Obtained also from the odd bisection of A254116.]
%o A254118 (PARI)
%o A254118 default(primelimit, 2^30);
%o A254118 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A254118 A064216(n) = A064989((2*n)-1);
%o A254118 A254103(n) = { if(0==n,0,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2)); };
%o A254118 A254116(n) = A064216(A254103(n));
%o A254118 A254118(n) = (A254116(n+n+1)-1)/2;
%o A254118 for(n=1, 8191, write("b254118.txt", n, " ", A254118(n)));
%o A254118 (Scheme, two versions)
%o A254118 (define (A254118 n) (+ -1 (A249745 (+ 1 (A254103 n)))))
%o A254118 (define (A254118 n) (/ (+ -1 (A254116 (+ 1 n n))) 2))
%o A254118 (Python)
%o A254118 from sympy import factorint, prevprime, floor
%o A254118 from operator import mul
%o A254118 from functools import reduce
%o A254118 def a064216(n):
%o A254118     f=factorint(2*n - 1)
%o A254118     return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f])
%o A254118 def a254103(n):
%o A254118     if n==0: return 0
%o A254118     if n%2==0: return 3*a254103(n//2) - 1
%o A254118     else: return floor((3*(1 + a254103((n - 1)/2)))//2)
%o A254118 def a254116(n): return a064216(a254103(n))
%o A254118 def a(n): return (a254116(2*n + 1) - 1)//2
%o A254118 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 06 2017
%Y A254118 Inverse: A254117.
%Y A254118 Other related permutations: A254116, A249745, A254103 (compare to the scatterplot of this one).
%Y A254118 Cf. A254120 (= a(2^n)).
%K A254118 nonn,look
%O A254118 1,2
%A A254118 _Antti Karttunen_, Feb 05 2015