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A342475
Prime numbers whose binary expansion contains only prime powers of 2 and the zeroth power.
1
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
1,1
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
EXAMPLE
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.
MATHEMATICA
Select[Array[1 + Total@ MapIndexed[#1*2^Prime[#2] & @@ {#1, First[#2]} &, Reverse@ IntegerDigits[#, 2]] &, 140], PrimeQ] (* Michael De Vlieger, Mar 13 2021 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
STATUS
approved