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%I #34 Sep 08 2022 08:46:10
%S 3,5,7,11,13,19,31,41,61,67,73,79,83,89,97,103,137,139,149,151,157,
%T 181,193,199,211,223,227,239,241,271,311,317,331,337,349,373,421,433,
%U 439,443,449,461,607,619,631,643,661,691,719,739,757,811,823,829,853,859
%N Primes of the form n^2 + k such that n^2 - k is also prime, where -n < k < n.
%C Members of a pair (a, b) of primes such that a < b and the distances from a and b to the nearest square above a (or below b) are equal.
%C The only prime of the form n^2 + 1 (A002496) in the sequence is 5.
%C Is this sequence infinite?
%F A prime p is in the sequence if and only if 2*A053187(p)-p is prime.
%e 2^2-1=3, 2^2+1=5, both prime.
%e 8^2-3=61, 8^2+3=67, both prime.
%p g:= proc(t,m) if isprime(m+t) and isprime(m-t) then (m+t,m-t) else NULL fi end proc:
%p `union`(seq(map(g,{$1..n-1},n^2),n=2..100));
%p # if using Maple 11 or earlier, uncomment the next line
%p # sort(convert(%,list));
%p # _Robert Israel_, Oct 31 2014
%o (Magma) lst:=[]; for m in [1..28] do r:=m*(m+1)+1; s:=(m+1)^2; for a in [r..s-1] do if IsPrime(a) then b:=2*s-a; if IsPrime(b) then Append(~lst, a); Append(~lst, b); end if; end if; end for; end for; Sort(lst);
%o (PARI) for(n=1, 859, if(issquare(n), x=ps=n; until(issquare(x), x++); ns=x); if(isprime(n), if(n-ps<ns-n, c=2*ps-n, c=2*ns-n); if(isprime(c), print1(n, ", "))));
%Y Cf. A047972, A053187.
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Oct 20 2014
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