A363537 - OEIS

%I #6 Jun 11 2023 12:29:33

%S 0,1,2,4,3,8,5,16,9,32,6,17,64,10,33,128,18,7,65,12,256,34,11,129,20,

%T 512,66,19,257,36,1024,13,130,24,35,513,68,2048,21,258,40,67,1025,132,

%U 4096,37,514,72,14,131,2049,25,260,48,8192,69,1026,136,22,259,4097,41,516,80,16384,133,2050,264,38

%N Rewrite A087980(n) = Product_{i=1..m} p(i)^e(i) instead as Sum_{i=1..m} 2^(i-1), where m = omega(A087980(n)) = A001221(A087980(n)).

%C Permutation of nonnegative numbers.

%C Rewriting nonnegative numbers n = Sum_{i=1..A000120(n)} 2^i instead as Product_{i=1..A000120(n)} p(i)^(e(i)+1) gives A362227.

%H Michael De Vlieger, <a href="/A363537/b363537.txt">Table of n, a(n) for n = 1..11157</a> (a(11157) = 2^64.)

%H Michael De Vlieger, <a href="/A363537/a363537.png">Log log scatterplot of a(n)</a>, n = 1..1203278.

%H Michael De Vlieger, <a href="/A363537/a363537_1.png">Plot S(n,k) in row a(n) of A272011 at (x,y) = (n,k)</a>, n = 1..2048.

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

%F a(A006939(k)) = 2^k-1 for k > 0.

%e Table relating this sequence to A087980, where b(n) = A087980(n), f(n) = A067255(n), g(n) = A272011(n), and a(n)_2 the binary expansion of a(n):

%e    n   b(n)  f(b(n))  a(n)  g(a(n))   a(n)_2

%e    1     1   0         0

%e    2     2   1         1    0             1

%e    3     4   2         2    1            1.

%e    4     8   3         4    2           1..

%e    5    12   2,1       3    1,0          11

%e    6    16   4         8    3          1...

%e    7    24   3,1       5    2,0         1.1

%e    8    32   5        16    4         1....

%e    9    48   4,1       9    3,0        1..1

%e   10    64   6        32    5        1.....

%e   11    72   3,2       6    2,1         11.

%e   12    96   5,1      17    4,0       1...1

%e   13   128   7        64    6       1......

%e   14   144   4,2      10    3,1        1.1.

%e   15   192   6,1      33    5,0      1....1

%e   16   256   8       128    7      1.......

%e   17   288   5,2      18    4,1       1..1.

%e   18   360   3,2,1     7    2,1,0       111

%e   ...

%t m = 15; f[n_] := Times @@ MapIndexed[Prime[First[#2]]^(#1 + 1) &, Length[#] - Position[#, 1][[All, 1]]] &[IntegerDigits[n, 2]]; SortBy[Select[Array[{#, f[#]} &, 2^(m + 1)], Last[#] <= 2^m &], Last][[All, 1]]

%Y Cf. A000079, A001221, A006939, A087980, A362227.

%K nonn

%O 1,3

%A _Michael De Vlieger_, Jun 09 2023