A305411 - OEIS

%I #59 Jan 11 2023 06:22:06

%S 3,5,17,37,43,67,109,151,157,163,181,199,229,331,367,373,421,433,577,

%T 601,613,691,757,823,919,997,1009,1051,1093,1231,1399,1429,1459,1579,

%U 1669,1777,1789,1831,2083,2179,2389,2521,2731,3019,3229,3301

%N Prime p numbers for which the sum with any prime number q <= p is not a square.

%H Robert Israel, <a href="/A305411/b305411.txt">Table of n, a(n) for n = 1..103</a> (conjectured to be all the terms)

%H C. Güllü et al, Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/4614952/primes-that-dont-give-a-square-when-added-any-of-the-primes-less-than-them/4614974#4614974">"Primes that don't give a square when added any of the primes less than them"</a>, Jan 2023.

%e 2 + 2 = 4, so 2 is not in the sequence.

%e 7 + 2 = 9, so 7 is not in the sequence.

%e 3 + 2 = 5 and 3 + 3 = 6. 5 and 6 are not squares, so 3 is a term.

%e 17 + 2 = 19, 17 + 3 = 20, 17 + 5 = 22, 17 + 7 = 24, 17 + 11 = 28, 17 + 13 = 30, 17 + 17 = 34. All these are nonsquares, so 17 is a term.

%p filter:= proc(p) local s, s0, q;

%p   if issqr(p+2) then return false fi;

%p   s0:= floor(sqrt(p));

%p   if s0::odd then s0:= s0-1 fi;

%p   for s from s0+2 by 2 do

%p      q:= s^2-p;

%p      if q > p then return true fi;

%p      if isprime(q) then return false fi;

%p   od;

%p end proc:

%p R:= NULL: p:= 1:

%p while p < 10^7 do

%p   p:= nextprime(p);

%p   if filter(p) then R:= R,p; fi;

%p od:

%p R; # _Robert Israel_, Jan 10 2023

%t aQ[p_] := PrimeQ[p] && Module[{q = 2}, While[q <= p && !IntegerQ[Sqrt[p + q]], q = NextPrime[q]]; q > p]; Select[Range[1000], aQ] (* _Amiram Eldar_, Dec 04 2018 *)

%o (PARI) isok(p) = {if (! isprime(p), return (0)); forprime(q=2, p, if (issquare(p+q), return (0));); return (1);} \\ _Michel Marcus_, Oct 18 2018

%Y Cf. A000040, A000290, A001043.

%K nonn

%O 1,1

%A _Marius A. Burtea_, Oct 13 2018