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A109306
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Numbers k such that k^2 + (k-1)^2 and k^2 + (k+1)^2 are both primes.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n)^2 = A075577(n). - David A. Corneth, Apr 25 2021
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EXAMPLE
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25 is a term because 25^2 + 24^2 = 1201 and 25^2 + 26^2 = 1301 are both primes.
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MATHEMATICA
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Select[Range[2, 10000], PrimeQ[ #^2+(#+1)^2]&&PrimeQ[ #^2+(#-1)^2]&]
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PROG
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(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
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CROSSREFS
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Cf. A027861, A075577.
Sequence in context: A326971 A000895 A351895 * A009560 A333591 A079434
Adjacent sequences: A109303 A109304 A109305 * A109307 A109308 A109309
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KEYWORD
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nonn
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AUTHOR
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Zak Seidov, Jun 25 2005
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EXTENSIONS
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Definition corrected by Walter Kehowski, Jul 04 2005
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STATUS
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approved
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