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

Author

Arkadiusz Wesolowski, Mar 17 2017

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000215, A109925.

Programs

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;

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 *)

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

Formula

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