A318400 Numbers whose prime indices are all powers of 2 (including 1).

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 19, 21, 24, 27, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 128, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 256, 262

Offset: 1

Author

Gus Wiseman, Dec 16 2018

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of all integer partitions whose parts are all powers of 2 (including 1) begins: (), (1), (2), (11), (21), (4), (111), (22), (211), (41), (1111), (221), (8), (42), (2111), (222), (411), (11111), (2211), (81), (421), (21111), (44).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003963, A018819, A033844, A056239, A106349, A112798, A302242, A302491, A322551.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pow2Q[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,_}}]];
Select[Range[100],And@@pow2Q/@primeMS[#]&]

Formula

Sum_{n>=1} 1/a(n) = 1/Product_{k>=0} (1 - 1/prime(2^k)) = 3.81625872357742992578... . - Amiram Eldar, Dec 03 2022