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

1,2

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.

Links

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

Formula

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

Example

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).

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[#]&]

Crossrefs

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

Sequence in context: A010408 A010455 A269804 * A219174 A108319 A177500

Adjacent sequences: A318397 A318398 A318399 * A318401 A318402 A318403

Keyword

nonn

Author

Gus Wiseman, Dec 16 2018

Status

approved