A342475 - OEIS

%I #30 May 09 2021 10:21:40

%S 5,13,37,41,137,173,2053,2081,2089,2213,2221,8233,8237,8329,8353,

%T 10253,10273,10369,131113,131213,133121,133153,133157,133253,133261,

%U 139273,139297,139301,139309,139393,139397,139429,141353,141481,524429,524453,526373,526381,526501

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

%C 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

%e 5 = 2^2 + 2^0 is a term.

%e 7 = 2^2 + 2^1 + 2^0 is not a term, because the exponent 1 is not a prime.

%e 11 = 2^3 + 2^1 + 2^0 is not a term, because the exponent 1 is not a prime.

%e 13 = 2^3 + 2^2 + 2^0 is a term.

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

%o (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

%Y Cf. A004676, A326782, A342481.

%K nonn,base

%O 1,1

%A _Vassilis Papadimitriou_, Mar 13 2021