A109306 Numbers k such that k^2 + (k-1)^2 and k^2 + (k+1)^2 are both primes.

2, 5, 25, 30, 35, 70, 85, 100, 110, 225, 230, 260, 285, 290, 320, 390, 410, 475, 490, 495, 515, 590, 680, 695, 710, 750, 760, 845, 950, 1080, 1100, 1135, 1175, 1190, 1195, 1270, 1295, 1305, 1330, 1365, 1410, 1475, 1715, 1750, 1785, 1845, 1855, 1925, 2015, 2060

Offset

1,1

Comments

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.

Links

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Formula

a(n)^2 = A075577(n). - David A. Corneth, Apr 25 2021

Example

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

Mathematica

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

Prog

(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
(Python)
from sympy import isprime
def aupto(limit):
  alst, is2 = [], False
  for k in range(1, limit+1):
    is1, is2 = is2, isprime(k**2 + (k+1)**2)
    if is1 and is2: alst.append(k)
  return alst
print(aupto(2060)) # Michael S. Branicky, Apr 25 2021

Crossrefs

Cf. A027861, A075577.

Sequence in context: A326971 A000895 A351895 * A009560 A333591 A079434

Adjacent sequences: A109303 A109304 A109305 * A109307 A109308 A109309

Keyword

nonn

Author

Zak Seidov, Jun 25 2005

Extensions

Definition corrected by Walter Kehowski, Jul 04 2005

Status

approved