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

0,4

Comments

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

Links

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

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A339612 A075257 A260618 * A226651 A073711 A071690

Adjacent sequences: A306283 A306284 A306285 * A306287 A306288 A306289

Keyword

nonn,base,easy

Author

Rémy Sigrist, May 04 2019

Status

approved