A207901 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k, also let R_k equal S_k read in reverse order; then the numbers b_k*R_k are the next 2^k terms.

1, 2, 6, 3, 12, 24, 8, 4, 20, 40, 120, 60, 15, 30, 10, 5, 35, 70, 210, 105, 420, 840, 280, 140, 28, 56, 168, 84, 21, 42, 14, 7, 63, 126, 378, 189, 756, 1512, 504, 252, 1260, 2520, 7560, 3780, 945, 1890, 630, 315, 45, 90, 270, 135, 540, 1080, 360, 180, 36, 72, 216

Offset: 0

Author

Paul D. Hanna, Feb 21 2012

Keywords

• nonn
• look

Comments

A permutation of the positive integers (but please note the starting offset: 0-indexed).

This sequence is a variant of A052330.

Shares with A064736, A302350, etc. the property that a(n) is either a divisor or a multiple of a(n+1). - Peter Munn, Apr 11 2018 on SeqFan-list. Note: A302781 is another such "divisor-or-multiple permutation" satisfying the same property. - Antti Karttunen, Apr 14 2018

The offset is 0 since S_0 = {1} denotes the first 2^0 = 1 terms. - Daniel Forgues, Apr 13 2018

This is "Fermi-Dirac piano played with Gray code", as indicated by Peter Munn's Apr 11 2018 formula. Compare also to A303771 and A302783. - Antti Karttunen, May 16 2018

Examples

			Start with [1]; appending 2*[1] results in [1,2];
appending 3*[2,1] results in [1,2, 6,3];
appending 4*[3,6,2,1] results in [1,2,6,3, 12,24,8,4];
appending 5*[4,8,24,12,3,6,2,1]
results in [1,2,6,3,12,24,8,4, 20,40,120,60,15,30,10,5];
next append 7*[5,10,30,15,60,120,40,20,4,8,24,12,3,6,2,1],
multiplying by 7 since 6 is already found in the previous terms.
Each new factor is in A050376: [2,3,4,5,7,9,11,13,16,17,19,23,25,29,...].
Continue in this way to generate all the terms of this sequence.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8191 (first 1024 terms from Paul D. Hanna)
• Michel Marcus, Peter Munn, et al, Discussion on SeqFan-list
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A001511, A003188, A052330, A050376, A059897, A064706, A302029 (inverse), A302843.

Cf. A064736, A281978, A282291, A302350, A302781, A302783, A303751, A303771, A304085, A304531, A304755 for other divisor-or-multiple permutations or conjectured permutations.

Cf. A302033 (a squarefree analog), A304745.

Programs

Mathematica

a = {1}; Do[a = Join[a, Reverse[a]*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

PARI

{A050376(n)= local(m, c, k, p); n--; if(n<=0, 2*(n==0), c=0; m=2; while( cA050376(n-1)*Vec(Polrev(A))));A[n]}
for(n=0,63,print1(a(n),",")) \\ edited for offsets by Michel Marcus, Apr 04 2019

PARI

up_to_e = 13;
v050376 = vector(up_to_e);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A003188(n) = bitxor(n, n>>1);
A207901(n) = A052330(A003188(n)); \\ Antti Karttunen, Apr 13 2018

Formula

a(n) = A052330(A003188(n)). - Peter Munn, Apr 11 2018

a(n) = A302781(A302843(n)) = A302783(A064706(n)). - Antti Karttunen, Apr 16 2018

a(n+1) = A059897(a(n), A050376(A001511(n+1))). - Peter Munn, Apr 01 2019

Extensions

Offset changed from 1 to 0 by Antti Karttunen, Apr 13 2018