A306286 a(n) is the product of the positions of the ones in the binary expansion of n (the most significant bit having position 1).

1, 1, 1, 2, 1, 3, 2, 6, 1, 4, 3, 12, 2, 8, 6, 24, 1, 5, 4, 20, 3, 15, 12, 60, 2, 10, 8, 40, 6, 30, 24, 120, 1, 6, 5, 30, 4, 24, 20, 120, 3, 18, 15, 90, 12, 72, 60, 360, 2, 12, 10, 60, 8, 48, 40, 240, 6, 36, 30, 180, 24, 144, 120, 720, 1, 7, 6, 42, 5, 35, 30

Offset: 0

Author

Rémy Sigrist, May 04 2019

Keywords

• nonn
• base
• easy

Comments

The variant where the least significant bit has position 1 corresponds to A096111 (with an appropriate offset).

Examples

			The first terms, alongside the positions of ones and the binary representation of n, are:
  n   a(n)  Pos. ones  bin(n)
  --  ----  ---------  ------
   0     1  {}              0
   1     1  {1}             1
   2     1  {1}            10
   3     2  {1,2}          11
   4     1  {1}           100
   5     3  {1,3}         101
   6     2  {1,2}         110
   7     6  {1,2,3}       111
   8     1  {1}          1000
   9     4  {1,4}        1001
  10     3  {1,3}        1010
  11    12  {1,3,4}      1011
  12     2  {1,2}        1100
  13     8  {1,2,4}      1101
  14     6  {1,2,3}      1110
  15    24  {1,2,3,4}    1111
  16     1  {1}         10000
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..16384

Crossrefs

Cf. A096111, A306549, A307218 (fixed points).

Programs

Mathematica

A306286[n_] := Times @@ Flatten[Position[IntegerDigits[n, 2], 1]];
Array[A306286, 100, 0] (* Paolo Xausa, Jun 01 2024 *)

PARI

a(n) = my (b=binary(n)); prod(k=1, #b, if (b[k],k,1))

PARI

a(n) = vecprod(Vec(select(x->(x==1), binary(n), 1))); \\ Michel Marcus, Jun 01 2024

Python

from math import prod
def a(n): return prod(i for i, bi in enumerate(bin(n)[2:], 1) if bi == "1")
print([a(n) for n in range(71)]) # Michael S. Branicky, Jun 01 2024

Formula

a(2*n) = a(n).

a(2^k) = 1 for any k >= 0.

a(2^k-1) = k! for any k >= 0.

a(2^k+1) = k+1 for any k >= 0.