A306286 - OEIS

%I #89 Jun 01 2024 14:46:25

%S 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,

%T 24,120,1,6,5,30,4,24,20,120,3,18,15,90,12,72,60,360,2,12,10,60,8,48,

%U 40,240,6,36,30,180,24,144,120,720,1,7,6,42,5,35,30

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

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

%H Rémy Sigrist, <a href="/A306286/b306286.txt">Table of n, a(n) for n = 0..16384</a>

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

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

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

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

%e The first terms, alongside the positions of ones and the binary representation of n, are:

%e   n   a(n)  Pos. ones  bin(n)

%e   --  ----  ---------  ------

%e    0     1  {}              0

%e    1     1  {1}             1

%e    2     1  {1}            10

%e    3     2  {1,2}          11

%e    4     1  {1}           100

%e    5     3  {1,3}         101

%e    6     2  {1,2}         110

%e    7     6  {1,2,3}       111

%e    8     1  {1}          1000

%e    9     4  {1,4}        1001

%e   10     3  {1,3}        1010

%e   11    12  {1,3,4}      1011

%e   12     2  {1,2}        1100

%e   13     8  {1,2,4}      1101

%e   14     6  {1,2,3}      1110

%e   15    24  {1,2,3,4}    1111

%e   16     1  {1}         10000

%t A306286[n_] := Times @@ Flatten[Position[IntegerDigits[n, 2], 1]];

%t Array[A306286, 100, 0] (* _Paolo Xausa_, Jun 01 2024 *)

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

%o (PARI) a(n) = vecprod(Vec(select(x->(x==1), binary(n), 1))); \\ _Michel Marcus_, Jun 01 2024

%o (Python)

%o from math import prod

%o def a(n): return prod(i for i, bi in enumerate(bin(n)[2:], 1) if bi == "1")

%o print([a(n) for n in range(71)]) # _Michael S. Branicky_, Jun 01 2024

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

%K nonn,base,easy

%O 0,4

%A _Rémy Sigrist_, May 04 2019