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A306549 a(n) is the product of the positions of the zeros in the binary expansion of n (the most significant bit having position 1).

%I A306549 #30 Jun 01 2024 15:05:14
%S A306549 1,1,2,1,6,2,3,1,24,6,8,2,12,3,4,1,120,24,30,6,40,8,10,2,60,12,15,3,
%T A306549 20,4,5,1,720,120,144,24,180,30,36,6,240,40,48,8,60,10,12,2,360,60,72,
%U A306549 12,90,15,18,3,120,20,24,4,30,5,6,1,5040,720,840,120,1008
%N A306549 a(n) is the product of the positions of the zeros in the binary expansion of n (the most significant bit having position 1).
%C A306549 Apparently, the variant where the least significant bit has position 1 corresponds to A124773.
%H A306549 Rémy Sigrist, <a href="/A306549/b306549.txt">Table of n, a(n) for n = 0..16384</a>
%F A306549 a(n) = A070939(n)! / A306286(n).
%F A306549 a(2*n) = a(n) * (1+A070939(n)).
%F A306549 a(2*n+1) = a(n).
%F A306549 a(2^k) = (k+1)! for any k >= 0.
%F A306549 a(2^k-1) = 1 for any k >= 0.
%F A306549 a(2^k-2) = k for any k >= 1.
%e A306549 The first terms, alongside the positions of zeros and the binary representation of n, are:
%e A306549   n   a(n)  Pos.zeros  bin(n)
%e A306549   --  ----  ---------  ------
%e A306549    0     1  {1}             0
%e A306549    1     1  {}              1
%e A306549    2     2  {2}            10
%e A306549    3     1  {}             11
%e A306549    4     6  {2,3}         100
%e A306549    5     2  {2}           101
%e A306549    6     3  {3}           110
%e A306549    7     1  {}            111
%e A306549    8    24  {2,3,4}      1000
%e A306549    9     6  {2,3}        1001
%e A306549   10     8  {2,4}        1010
%e A306549   11     2  {2}          1011
%e A306549   12    12  {3,4}        1100
%e A306549   13     3  {3}          1101
%e A306549   14     4  {4}          1110
%e A306549   15     1  {}           1111
%t A306549 A306549[n_] := Times @@ Flatten[Position[IntegerDigits[n, 2], 0]];
%t A306549 Array[A306549, 100, 0] (* _Paolo Xausa_, Jun 01 2024 *)
%o A306549 (PARI) a(n) = my (b=binary(n)); prod(k=1, #b, if (b[k]==0, k, 1))
%o A306549 (PARI) a(n) = vecprod(Vec(select(x->(x==0), binary(n), 1)));
%o A306549 (Python)
%o A306549 from math import prod
%o A306549 def a(n): return prod(i for i, bi in enumerate(bin(n)[2:], 1) if bi == "0")
%o A306549 print([a(n) for n in range(70)]) # _Michael S. Branicky_, Jun 01 2024
%Y A306549 Cf. A070939, A124773, A306286.
%K A306549 nonn,base,easy
%O A306549 0,3
%A A306549 _Rémy Sigrist_, May 04 2019