A283806 Odd numbers which are uniquely decomposable into the sum of a prime and a power of two.

3, 5, 17, 29, 41, 53, 59, 65, 89, 97, 119, 137, 163, 179, 185, 191, 193, 209, 217, 219, 221, 223, 233, 239, 247, 253, 269, 281, 305, 307, 311, 343, 359, 389, 403, 407, 415, 419, 427, 431, 457, 491, 505, 521, 533, 545, 547, 557, 569, 575, 581, 583, 597, 613, 637

Offset

1,1

Comments

It is conjectured that none of these numbers is in A101036.

A positive integer n belongs to this sequence if n is of the form x*y + x - 1 and for some m >= 1:

1) y = -1 + 2 * Product_{k=0..m} (2^(2^k) + 1),

2) x <= 2^(2^(m+1) - 1),

3) n - 2^(2^(m+1)) is prime.

Odd numbers m that satisfy A109925(m) = 1. - Michel Marcus, Mar 19 2017

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Wikipedia, Prime number

Formula

a(n) ~ 10*(n + n/log(n)).

Example

17 is in the sequence since 17 - 2^2 = 13 is a prime and 17 - 2^0 = 16, 17 - 2^1 = 15, 17 - 2^3 = 9, 17 - 2^4 = 1 are all nonprimes.

Mathematica

Select[Range[1, 640, 2], Function[n, Total@ Boole@ PrimeQ@ Map[n - # &, 2^Range[0, Floor@ Log2@ n]] == 1]] (* Michael De Vlieger, Mar 18 2017 *)

Prog

(Magma) lst:=[]; for n in [1..637 by 2] do c:=0; r:=Floor(Log(n)/Log(2)); for x in [0..r] do a:=n-2^x; if IsPrime(a) then c+:=1; end if; if c eq 2 then break; end if; end for; if c eq 1 then Append(~lst, n); end if; end for; lst;
(PARI) isok(n) = (n % 2) && (sum(k=0, log(n)\log(2), isprime(n-2^k)) == 1); \\ Michel Marcus, Mar 18 2017
(Python)
from sympy import isprime
import math
print([n for n in range(1001) if n%2 and sum([isprime(n-2**k) for k in range(int(math.floor(math.log(n)/math.log(2))) + 1)]) == 1]) # Indranil Ghosh, Mar 29 2017

Crossrefs

Cf. A000215, A109925.

Sequence in context: A079373 A181291 A038703 * A163586 A074931 A023226

Adjacent sequences: A283803 A283804 A283805 * A283807 A283808 A283809

Keyword

nonn,easy

Author

Arkadiusz Wesolowski, Mar 17 2017

Status

approved