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

1,2

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)

Formula

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

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 *)

Prog

(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

Crossrefs

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

Positions of 1's in A336467.

Subsequences: A054784, A335431.

Sequence in context: A010455 A269804 A318400 * A108319 A177500 A018772

Adjacent sequences: A219171 A219172 A219173 * A219175 A219176 A219177

Keyword

nonn

Author

Jon Perry, Nov 13 2012

Extensions

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

Status

approved