A219174 Numbers that have no other prime factors than 2 and/or Mersenne primes.

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 27, 28, 31, 32, 36, 42, 48, 49, 54, 56, 62, 63, 64, 72, 81, 84, 93, 96, 98, 108, 112, 124, 126, 127, 128, 144, 147, 162, 168, 186, 189, 192, 196, 216, 217, 224, 243, 248, 252, 254, 256, 279, 288, 294, 324, 336, 343, 372

Offset: 1

Author

Jon Perry, Nov 13 2012

Keywords

• nonn

Comments

If k is in the sequence, then so is 2*k.

From Antti Karttunen, Jul 16 2023: (Start)

The original definition was "Numbers whose prime factors are either 2 or Mersenne primes". The new definition admits also {1}.

Multiplicative semigroup. Primitive terms are {1, 2} U A000668.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001 (original 10000 initial terms from Amiram Eldar)

Crossrefs

Cf. A000668, A056652, A108319, A364252 (characteristic function).

Positions of 1's in A336467.

Subsequences: A054784, A335431.

Programs

Mathematica

seq[max_] := Module[{e = Floor @ Log2[max + 1], s = {1}, es, ps, n, p, m}, es = Select[MersennePrimeExponent @ Range[20], # <= e &]; ps = Join[{2}, 2^es - 1];  n = Length[ps]; Do[p = ps[[k]]; m = Floor @ Log[p, max]; s = Select[Union @ Flatten@Outer[Times, s, p^Range[0, m]], # <= max &], {k, 1, n}]; s]; seq[10^3] (* Amiram Eldar, Jan 09 2021 *)

PARI

isokp(p) = my(q); (p==2) || (isprimepower(p+1, &q) && (q==2));
isok(m) = ((1==m) || vecmin(apply(isokp, factor(m)[, 1]))); \\ Michel Marcus, Jan 09 2021, edited by Antti Karttunen, Jul 16 2023

PARI

isok(n) = A364252(n); \\ Antti Karttunen, Jul 16 2023

Formula

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} p/(p-1) = 3.6458502419452069302... - Amiram Eldar, Jan 09 2021

Extensions

a(1) = 1 prepended, and definition changed accordingly by Antti Karttunen, Jul 16 2023