A075577 -id:A075577 - cp's OEIS frontend

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

Zak Seidov, Jun 25 2005

nonn

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.

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

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

Cf. A027861, A075577.

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

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

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

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

Definition corrected by Walter Kehowski, Jul 04 2005

A176796 Numbers k such that A129307(k) + A129307(k+1) is a square.

Original entry on oeis.org

1, 3, 12, 14, 17, 25, 30, 35, 39, 69, 71, 74, 80, 83, 88, 102, 107, 122, 126, 129, 134, 151, 170, 172, 176, 184, 187, 202, 220, 239, 244, 249, 258, 261, 263, 272, 280, 283, 289, 298, 308, 321, 363, 371, 377, 386, 390, 403, 421, 432, 438, 447, 451, 453, 477, 480

Offset: 1

Giovanni Teofilatto, Apr 26 2010

nonn

Squares are in A075577.

Cf. A000217, A027861, A075577, A129307.

A027861 := proc(n) option remember; local a; if n= 1 then 1; else for a from procname(n-1)+1 do if isprime(a^2+(a+1)^2) then return a; end if; end do: end if; end proc:
A129307 := proc(n) A000217(A027861(n)) ; end proc:
A176796 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if issqr(A129307(a)+A129307(a+1)) then return a; end if; end do: end if; end proc:
seq(A176796(n),n=1..80) ; # R. J. Mathar, Jun 28 2010

Extended beyond a(5) by R. J. Mathar, Jun 28 2010

cp's OEIS Frontend

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

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