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%I #44 Sep 08 2022 08:46:19
%S 3,5,17,29,41,53,59,65,89,97,119,137,163,179,185,191,193,209,217,219,
%T 221,223,233,239,247,253,269,281,305,307,311,343,359,389,403,407,415,
%U 419,427,431,457,491,505,521,533,545,547,557,569,575,581,583,597,613,637
%N Odd numbers which are uniquely decomposable into the sum of a prime and a power of two.
%C It is conjectured that none of these numbers is in A101036.
%C A positive integer n belongs to this sequence if n is of the form x*y + x - 1 and for some m >= 1:
%C 1) y = -1 + 2 * Product_{k=0..m} (2^(2^k) + 1),
%C 2) x <= 2^(2^(m+1) - 1),
%C 3) n - 2^(2^(m+1)) is prime.
%C Odd numbers m that satisfy A109925(m) = 1. - _Michel Marcus_, Mar 19 2017
%H Robert G. Wilson v, <a href="/A283806/b283806.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_number">Prime number</a>
%F a(n) ~ 10*(n + n/log(n)).
%e 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.
%t 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 *)
%o (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;
%o (PARI) isok(n) = (n % 2) && (sum(k=0, log(n)\log(2), isprime(n-2^k)) == 1); \\ _Michel Marcus_, Mar 18 2017
%o (Python)
%o from sympy import isprime
%o import math
%o 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
%Y Cf. A000215, A109925.
%K nonn,easy
%O 1,1
%A _Arkadiusz Wesolowski_, Mar 17 2017
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