A342475 Prime numbers whose binary expansion contains only prime powers of 2 and the zeroth power.

5, 13, 37, 41, 137, 173, 2053, 2081, 2089, 2213, 2221, 8233, 8237, 8329, 8353, 10253, 10273, 10369, 131113, 131213, 133121, 133153, 133157, 133253, 133261, 139273, 139297, 139301, 139309, 139393, 139397, 139429, 141353, 141481, 524429, 524453, 526373, 526381, 526501

Offset: 1

Author

Vassilis Papadimitriou, Mar 13 2021

Keywords

• nonn
• base

Comments

The numbers m = 2^e(0) + 2^e(1) + 2^e(2) + ... where all e(i) are either 0 or prime are 1, 4, 5, 8, 9, 12, 13, 32, 33, 36, 37, 40, 41, 44, 45, 128, 129, 132, 133, 136, 137, 140, 141, 160, 161, 164, ... The sequence contains the m which are primes. - R. J. Mathar, Apr 21 2021

Examples

			5 = 2^2 + 2^0 is a term.
7 = 2^2 + 2^1 + 2^0 is not a term, because the exponent 1 is not a prime.
11 = 2^3 + 2^1 + 2^0 is not a term, because the exponent 1 is not a prime.
13 = 2^3 + 2^2 + 2^0 is a term.
		

Crossrefs

Cf. A004676, A326782, A342481.

Programs

Mathematica

Select[Array[1 + Total@ MapIndexed[#1*2^Prime[#2] & @@ {#1, First[#2]} &, Reverse@ IntegerDigits[#, 2]] &, 140], PrimeQ] (* Michael De Vlieger, Mar 13 2021 *)

PARI

isok(p) = if (isprime(p), my(b=Vecrev(binary(p))); sum(i=1, #b, b[i]*((i!=1) && !isprime(i-1))) == 0); \\ Michel Marcus, Apr 22 2021