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).

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, 30, 24, 45, 21, 20, 14, 11, 121, 98, 100, 147, 225, 72, 150, 245, 625, 162, 64, 243, 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

Offset: 0

Author

Antti Karttunen, Jun 05 2014

Keywords

• nonn

Comments

Note the indexing: the domain includes zero, but the range starts from one.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

A243354 gives the inverse mapping.

Cf. A227739, A112798, A075157, A241909, A005940, A003188, A226062, A242424.

Programs

Mathematica

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 *)

Python

from sympy import prime
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n)) # Indranil Ghosh, May 07 2017

Scheme

(define (A243353 n) (A005940 (+ 1 (A003188 n))))

Formula

a(n) = A005940(1+A003188(n)).

a(n) = A241909(1+A075157(n)). [With A075157's original starting offset]

For all n >= 0, A243354(a(n)) = n.

A227183(n) = A056239(a(n)). [Maps between the corresponding sums ...]

A227184(n) = A003963(a(n)). [... and products of parts of each partition].

For n >= 0, a(A037481(n)) = A002110(n). [Also "triangular partitions", the fixed points of Bulgarian solitaire, A226062 & A242424].

For n >= 1, a(A227451(n+1)) = 4*A243054(n).