A363537 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)).

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

Offset

1,3

Comments

Permutation of nonnegative numbers.

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11157 (a(11157) = 2^64.)

Michael De Vlieger, Log log scatterplot of a(n), n = 1..1203278.

Michael De Vlieger, Plot S(n,k) in row a(n) of A272011 at (x,y) = (n,k), n = 1..2048.

Formula

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

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

Example

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):
   n   b(n)  f(b(n))  a(n)  g(a(n))   a(n)_2
   1     1   0         0
   2     2   1         1    0             1
   3     4   2         2    1            1.
   4     8   3         4    2           1..
   5    12   2,1       3    1,0          11
   6    16   4         8    3          1...
   7    24   3,1       5    2,0         1.1
   8    32   5        16    4         1....
   9    48   4,1       9    3,0        1..1
  10    64   6        32    5        1.....
  11    72   3,2       6    2,1         11.
  12    96   5,1      17    4,0       1...1
  13   128   7        64    6       1......
  14   144   4,2      10    3,1        1.1.
  15   192   6,1      33    5,0      1....1
  16   256   8       128    7      1.......
  17   288   5,2      18    4,1       1..1.
  18   360   3,2,1     7    2,1,0       111
  ...

Mathematica

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]]

Crossrefs

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

Sequence in context: A355436 A352713 A054427 * A232563 A048672 A277517

Adjacent sequences: A363534 A363535 A363536 * A363538 A363539 A363540

Keyword

nonn

Author

Michael De Vlieger, Jun 09 2023

Status

approved