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

%I #7 May 12 2024 11:17:51

%S 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,

%T 448,10,104,512,25,1024,11,896,1792,3584,23,2048,7168,14336,38,4096,

%U 50,8192,208,416,28672,16384,31,24,832,57344,1664,32768,76,114688

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

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

%F 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).

%t z = 100; prisigs = Table[If[n == 1, {0}, Sort[Last /@ FactorInteger[n]]], {n, z}];

%t exps = Table[Count[Take[prisigs, n], prisigs[[n]]], {n, z}]; (*A064839*)

%t Table[2^(exps[[n]] - 1) (Times @@ Prime[1 + prisigs[[n]]] - 2), {n, z}]

%Y Cf. A064839, A181819, A372134.

%K nonn

%O 1,3

%A _Friedjof Tellkamp_, Apr 20 2024