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A236461 Sum of two consecutive primes that is also sum of two consecutive even positive squares.

%I A236461 #14 Jul 03 2021 16:08:30
%S A236461 52,100,340,1460,2452,2740,4420,20404,21220,36452,48052,62660,66980,
%T A236461 94180,103060,108580,128020,140452,142580,169364,171700,195940,221780,
%U A236461 254900,260644,361252,378452,490052,498004,717604,736900,756452,766324,791284,879140,889780,916660,1016740,1104100,1164340,1232452,1283204
%N A236461 Sum of two consecutive primes that is also sum of two consecutive even positive squares.
%C A236461 All values of (q - p) are multiples of 6.
%C A236461 m = p + q = x^2 + (x+2)^2; {m,p,q,x}: {52, 23, 29, 4}, {100, 47, 53, 6}, {340, 167,  173, 12}, {1460, 727, 733, 26}, {2452, 1223,  1229, 34}, {2740, 1367, 1373, 36}, {4420, 2207, 2213, 46}.
%C A236461 Intersection of A001043 and A108099. - _Michel Marcus_, Jan 27 2014
%H A236461 Robert Israel, <a href="/A236461/b236461.txt">Table of n, a(n) for n = 1..10000</a>
%e A236461 52 = 23 + 29 = 4^2 + 6^2.
%p A236461 count:= 0: R:= NULL:
%p A236461 for m from 1 while count < 100 do
%p A236461   y:= 8*m^2+8*m+4;
%p A236461   if prevprime(y/2) + nextprime(y/2)=y then
%p A236461      count:= count+1;
%p A236461      R:= R, y;
%p A236461   fi
%p A236461 od:
%p A236461 R; # _Robert Israel_, Jan 07 2020
%t A236461 With[{nn=100000},Intersection[Total/@Partition[Prime[Range[nn]],2,1],Total/@ Partition[Range[2,2nn,2]^2,2,1]]] (* _Harvey P. Dale_, Jul 03 2021 *)
%o A236461 (PARI) v=vector(1300000); pp=3; forprime(p=5,#v/2,v[p+pp]++;pp=p);forstep(k=2,sqrtint(#v/2)-1,2,v[2*(k^2+2*k+2)]++);for(k=1,#v,if(v[k]==2,print1(k,", "))) \\ _Hugo Pfoertner_, Jan 07 2020
%Y A236461 Cf. A001043, A108099.
%K A236461 nonn
%O A236461 1,1
%A A236461 _Zak Seidov_, Jan 26 2014