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

Author

Zak Seidov, Jun 25 2005

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A027861, A075577.

Programs

Mathematica

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

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

Formula

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

Extensions

Definition corrected by Walter Kehowski, Jul 04 2005