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

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, 95, 12, 39, 1023, 21, 2047, 16, 55, 191, 29, 18, 4095, 383, 111, 35, 8191, 43, 16383, 79, 25, 767, 32767, 33, 30, 22, 223, 159, 65535, 20, 59, 71, 447

Offset: 1

Author

Antti Karttunen, Jan 03 2015

Keywords

• nonn

Comments

Note the indexing.

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Antti Karttunen)
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A253563.

Cf. A054429, A122111, A156552, A161919, A253566.

Programs

Maple

a:= proc(n) local i, l, r; r, l:= 0, [0, sort(map(i->
      numtheory[pi](i[1])$i[2], ifactors(n)[2]))[]];
      for i to nops(l)-1 do
        r:= 2*((x-> 2*x+1)@@(l[i+1]-l[i]))(r)
      od; r/2
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jul 21 2017

Mathematica

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 *)

Scheme

(define (A253564 n) (A156552 (A122111 n)))

Formula

a(n) = A156552(A122111(n)).

As a composition of other permutations:

a(n) = A054429(A253566(n)).