A219174 - OEIS

%I #33 Jul 16 2023 10:56:50

%S 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,

%T 63,64,72,81,84,93,96,98,108,112,124,126,127,128,144,147,162,168,186,

%U 189,192,196,216,217,224,243,248,252,254,256,279,288,294,324,336,343,372

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

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

%C From _Antti Karttunen_, Jul 16 2023: (Start)

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

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

%C (End)

%H Antti Karttunen, <a href="/A219174/b219174.txt">Table of n, a(n) for n = 1..10001</a> (original 10000 initial terms from Amiram Eldar)

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

%t 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 *)

%o (PARI) isokp(p) = my(q); (p==2) || (isprimepower(p+1, &q) && (q==2));

%o isok(m) = ((1==m) || vecmin(apply(isokp, factor(m)[, 1]))); \\ _Michel Marcus_, Jan 09 2021, edited by _Antti Karttunen_, Jul 16 2023

%o (PARI) isok(n) = A364252(n); \\ _Antti Karttunen_, Jul 16 2023

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

%Y Positions of 1's in A336467.

%Y Subsequences: A054784, A335431.

%K nonn

%O 1,2

%A _Jon Perry_, Nov 13 2012

%E a(1) = 1 prepended, and definition changed accordingly by _Antti Karttunen_, Jul 16 2023