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A075577 k^2 is a term if k^2 + (k-1)^2 and k^2 + (k+1)^2 are primes.

%I A075577 #20 Apr 26 2021 01:48:18
%S A075577 4,25,625,900,1225,4900,7225,10000,12100,50625,52900,67600,81225,
%T A075577 84100,102400,152100,168100,225625,240100,245025,265225,348100,462400,
%U A075577 483025,504100,562500,577600,714025,902500,1166400,1210000,1288225,1380625,1416100,1428025
%N A075577 k^2 is a term if k^2 + (k-1)^2 and k^2 + (k+1)^2 are primes.
%C A075577 For a(2) onwards, a(n) == 0 (mod 25).
%H A075577 Michael S. Branicky, <a href="/A075577/b075577.txt">Table of n, a(n) for n = 1..10000</a>
%F A075577 a(n) = A109306(n)^2. - _David A. Corneth_, Apr 25 2021
%e A075577 900 = 30^2 is a term because 30^2 + 29^2 = 1741 is prime and 30^2 + 31^2 = 1861 is prime.
%t A075577 Do[s=n^2+(n-1)^2; s1=n^2+(n+1)^2; If[PrimeQ[s]&&PrimeQ[s1], Print[n^2]], {n, 1, 5000}]
%o A075577 (Python)
%o A075577 from sympy import isprime
%o A075577 def aupto(limit):
%o A075577   alst, is2 = [], False
%o A075577   for k in range(1, int(limit**.5) + 2):
%o A075577     is1, is2 = is2, isprime(k**2 + (k+1)**2)
%o A075577     if is1 and is2: alst.append(k**2)
%o A075577   return alst
%o A075577 print(aupto(1500000)) # _Michael S. Branicky_, Apr 25 2021
%Y A075577 Cf. A109306.
%K A075577 nonn
%O A075577 1,1
%A A075577 _Amarnath Murthy_, Sep 25 2002
%E A075577 More terms from _Labos Elemer_, Sep 27 2002
%E A075577 a(34) and beyond from _Michael S. Branicky_, Apr 25 2021