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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).

%I A306286 #89 Jun 01 2024 14:46:25
%S A306286 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 A306286 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 A306286 40,240,6,36,30,180,24,144,120,720,1,7,6,42,5,35,30
%N 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).
%C A306286 The variant where the least significant bit has position 1 corresponds to A096111 (with an appropriate offset).
%H A306286 Rémy Sigrist, <a href="/A306286/b306286.txt">Table of n, a(n) for n = 0..16384</a>
%F A306286 a(2*n) = a(n).
%F A306286 a(2^k) = 1 for any k >= 0.
%F A306286 a(2^k-1) = k! for any k >= 0.
%F A306286 a(2^k+1) = k+1 for any k >= 0.
%e A306286 The first terms, alongside the positions of ones and the binary representation of n, are:
%e A306286   n   a(n)  Pos. ones  bin(n)
%e A306286   --  ----  ---------  ------
%e A306286    0     1  {}              0
%e A306286    1     1  {1}             1
%e A306286    2     1  {1}            10
%e A306286    3     2  {1,2}          11
%e A306286    4     1  {1}           100
%e A306286    5     3  {1,3}         101
%e A306286    6     2  {1,2}         110
%e A306286    7     6  {1,2,3}       111
%e A306286    8     1  {1}          1000
%e A306286    9     4  {1,4}        1001
%e A306286   10     3  {1,3}        1010
%e A306286   11    12  {1,3,4}      1011
%e A306286   12     2  {1,2}        1100
%e A306286   13     8  {1,2,4}      1101
%e A306286   14     6  {1,2,3}      1110
%e A306286   15    24  {1,2,3,4}    1111
%e A306286   16     1  {1}         10000
%t A306286 A306286[n_] := Times @@ Flatten[Position[IntegerDigits[n, 2], 1]];
%t A306286 Array[A306286, 100, 0] (* _Paolo Xausa_, Jun 01 2024 *)
%o A306286 (PARI) a(n) = my (b=binary(n)); prod(k=1, #b, if (b[k],k,1))
%o A306286 (PARI) a(n) = vecprod(Vec(select(x->(x==1), binary(n), 1))); \\ _Michel Marcus_, Jun 01 2024
%o A306286 (Python)
%o A306286 from math import prod
%o A306286 def a(n): return prod(i for i, bi in enumerate(bin(n)[2:], 1) if bi == "1")
%o A306286 print([a(n) for n in range(71)]) # _Michael S. Branicky_, Jun 01 2024
%Y A306286 Cf. A096111, A306549, A307218 (fixed points).
%K A306286 nonn,base,easy
%O A306286 0,4
%A A306286 _Rémy Sigrist_, May 04 2019