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A187797 Numbers having at least two different ordered partitions p+q and (p+2)+(q-2) where p, q, p+2 and q-2 are all prime.

%I A187797 #26 Feb 01 2021 18:29:49
%S A187797 10,16,18,22,24,30,34,36,42,46,48,54,60,64,66,72,76,78,84,90,102,106,
%T A187797 108,112,114,120,126,132,138,142,144,150,154,156,162,168,174,180,184,
%U A187797 186,192,196,198,202,204,210,216,222,228,232,234,240,244,246,252,258,264,270,274,276,282,286
%N A187797 Numbers having at least two different ordered partitions p+q and (p+2)+(q-2) where p, q, p+2 and q-2 are all prime.
%C A187797 Numbers k with at least one pair of externally tangent circles with radius sqrt(2) and center (p,q) where p and q are prime, p + q = k and p <= q. - _Wesley Ivan Hurt_, Aug 11 2020
%e A187797 For n=10, the partition solutions are 3+7 and 5+5, giving p=3, q=7, p+2=5, q-2=5.
%p A187797 isA187797 := proc(n)
%p A187797     local i,p,q ;
%p A187797     for i from 1 do
%p A187797         p := ithprime(i) ;
%p A187797         q := n-p ;
%p A187797         if q <= p+2 then
%p A187797             return false;
%p A187797         end if;
%p A187797         if isprime(q) then
%p A187797             if isprime(p+2) and isprime(q-2) then
%p A187797                 return true;
%p A187797             end if;
%p A187797         end if;
%p A187797     end do:
%p A187797     return false;
%p A187797 end proc:
%p A187797 for n from 4 to 600 do
%p A187797     if isA187797(n) then
%p A187797         printf("%d,",n);
%p A187797     end if;
%p A187797 end do: # _R. J. Mathar_, Oct 03 2013
%t A187797 Table[If[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]) (PrimePi[i + 2] - PrimePi[i + 1]) (PrimePi[2 n - i - 2] - PrimePi[2 n - i - 3]), {i, n - 2}] > 0, 2 n, {}], {n, 100}] // Flatten (* _Wesley Ivan Hurt_, Apr 13 2020 *)
%K A187797 nonn
%O A187797 1,1
%A A187797 _Bob Gilson_, Aug 30 2013