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%I #8 Aug 02 2024 16:43:31
%S 5,17,37,61,73,89,101,109,113,149,181,193,197,257,277,281,317,337,349,
%T 373,401,409,449,509,521,569,577,601,613,653,677,701,709,757,761,773,
%U 797,809,937,953,1009,1021,1033,1049,1069,1117,1181,1193,1237,1249,1277,1297,1301
%N Prime numbers that are the sum of an odd square s and an even square t such that 0 < s < t.
%e 17 is in the sequence since it is a prime such that 1^2 + 4^2 = 1 + 16 = 17, where 1 is odd, 16 is even, and 0 < 1 < 16.
%t Table[If[(PrimePi[n] - PrimePi[n - 1])*Sum[Mod[i, 2] Mod[n - i + 1, 2] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 1200}] // Flatten
%t osesQ[p_]:=Module[{s=Select[IntegerPartitions[p,{2}],AllTrue[Sqrt[#],IntegerQ]&]},OddQ[ Min[s]] && EvenQ[Max[s]]]; Select[Prime[Range[250]],osesQ] (* _Harvey P. Dale_, Aug 02 2024 *)
%Y Cf. A010051, A010052.
%K nonn
%O 1,1
%A _Wesley Ivan Hurt_, Dec 25 2020