A372133 Prime shadow (A181819) with shifted e_i to generate odd numbers, multiplied by 2^(A064839 - 1).

0, 1, 2, 3, 4, 7, 8, 5, 6, 14, 16, 13, 32, 28, 56, 9, 64, 26, 128, 52, 112, 224, 256, 19, 12, 448, 10, 104, 512, 25, 1024, 11, 896, 1792, 3584, 23, 2048, 7168, 14336, 38, 4096, 50, 8192, 208, 416, 28672, 16384, 31, 24, 832, 57344, 1664, 32768, 76, 114688

Offset: 1

Author

Friedjof Tellkamp, Apr 20 2024

Keywords

• nonn

Comments

a(n>1) contains all positive integers exactly once.

Crossrefs

Cf. A064839, A181819, A372134.

Programs

Mathematica

z = 100; prisigs = Table[If[n == 1, {0}, Sort[Last /@ FactorInteger[n]]], {n, z}];
exps = Table[Count[Take[prisigs, n], prisigs[[n]]], {n, z}]; (*A064839*)
Table[2^(exps[[n]] - 1) (Times @@ Prime[1 + prisigs[[n]]] - 2), {n, z}]

Formula

Factor n, multiplying out prime(e_i + 1) generates all odd numbers >= 3. Subtract 2 and multiply by 2^(k - 1), where k is the number of occurrences of the same prime signature of n in (1..n) (A064839).