A109306 - OEIS

%I #23 Apr 26 2021 01:49:49

%S 2,5,25,30,35,70,85,100,110,225,230,260,285,290,320,390,410,475,490,

%T 495,515,590,680,695,710,750,760,845,950,1080,1100,1135,1175,1190,

%U 1195,1270,1295,1305,1330,1365,1410,1475,1715,1750,1785,1845,1855,1925,2015,2060

%N Numbers k such that k^2 + (k-1)^2 and k^2 + (k+1)^2 are both primes.

%C All terms, except for the first one, are multiples of 5. All corresponding primes, except the first, end in 1. Cf. A027861, where in pairs of successive numbers the larger one is a multiple of 5 and is a term in this sequence.

%H Daniel Starodubtsev, <a href="/A109306/b109306.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n)^2 = A075577(n). - _David A. Corneth_, Apr 25 2021

%e 25 is a term because 25^2 + 24^2 = 1201 and 25^2 + 26^2 = 1301 are both primes.

%t Select[Range[2, 10000], PrimeQ[ #^2+(#+1)^2]&&PrimeQ[ #^2+(#-1)^2]&]

%o (PARI) for(k=1,2060,my(j=2*k^2+1);if(isprime(j-2*k)&&isprime(j+2*k),print1(k,", "))) \\ _Hugo Pfoertner_, Dec 07 2019

%o (Python)

%o from sympy import isprime

%o def aupto(limit):

%o   alst, is2 = [], False

%o   for k in range(1, limit+1):

%o     is1, is2 = is2, isprime(k**2 + (k+1)**2)

%o     if is1 and is2: alst.append(k)

%o   return alst

%o print(aupto(2060)) # _Michael S. Branicky_, Apr 25 2021

%Y Cf. A027861, A075577.

%K nonn

%O 1,1

%A _Zak Seidov_, Jun 25 2005

%E Definition corrected by _Walter Kehowski_, Jul 04 2005