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%I #23 Jun 11 2016 12:40:26
%S 3,3,5,5,7,7,11,11,13,13,13,13,17,17,17,17,19,19,19,23,23,23,23,29,29,
%T 29,29,31,31,31,31,31,31,37,37,37,37,37,37,41,41,41,41,41,41,41,43,43,
%U 43,43,43,43,43,47,47,47,47,47,47
%N Difference between the closest squares surrounding prime p is prime.
%C Conjecture: The number of terms in this sequence is infinite.
%C The number of occurrences of odd prime x is A014085((x-1)/2). - _Robert Israel_, Jun 08 2016
%H Robert Israel, <a href="/A111213/b111213.txt">Table of n, a(n) for n = 2..10001</a>
%F Let p be a prime number and r = floor(sqrt(p)). Then the closest surrounding squares of p are r^2 and (r+1)^2. So d = (r+1)^2 - r^2 = 2r+1. If d is prime then list d.
%e 29 is a prime number. 5^2 and 6^2 are the closest squares surrounding 29. Now the difference, 36-25 = 11 is prime so 11 is in the table.
%p g:= proc(q) local x; x:= (q-1)/2; numtheory:-pi((x+1)^2) - numtheory:-pi(x^2) end proc:
%p seq(p$g(p), p = select(isprime,[seq(i,i=3..1000,2)])); # _Robert Israel_, Jun 08 2016
%t Select[Table[Ceiling[#]^2 - Floor[#]^2 &@ Sqrt@ Prime@ n, {n, 120}], PrimeQ] (* _Michael De Vlieger_, Jun 10 2016 *)
%o (PARI) surrsqpr(n) = { local(x,y,j,r,d); forprime(x=2,n, r=floor(sqrt(x)); d=r+r+1; if(isprime(d), print1(d",") ) ) }
%Y Cf. A014085.
%K easy,nonn
%O 2,1
%A _Cino Hilliard_, Nov 12 2005
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