A284211 -id:A284211 - cp's OEIS frontend

A284346 a(n) is the least positive integer such that n^2 + a(n)^2 and (n + 1)^2 + (a(n) + 1)^2 are primes.

2, 1, 8, 1, 4, 1, 2, 3, 16, 3, 6, 7, 8, 1, 4, 1, 22, 5, 6, 3, 4, 17, 18, 5, 4, 1, 32, 5, 10, 29, 4, 27, 8, 15, 18, 1, 2, 15, 10, 3, 4, 247, 8, 15, 14, 19, 22, 35, 6, 19, 4, 27, 10, 11, 8, 1, 2, 5, 40, 13, 44, 127, 58, 61, 28, 1, 22, 13, 10, 19, 6, 7, 8, 15, 4, 9

Offset: 1

Lars-Erik Svahn, Mar 25 2017

nonn

n is odd iff a(n) is even.

			a(1)=2 since (1 + 1)^2 + (1 + 1)^2 is not prime, but 1^2 + 2^2 = 5 and (1 + 1)^2 + (2 + 1)^2 = 13 are prime.

Lars-Erik Svahn, Table of n, a(n) for n = 1..10000
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.
Eric Weisstein's World of Mathematics, Gaussian Prime.
Index entries for Gaussian integers and primes.

Cf. A069003, A284211, A284327.

Rest@ FoldList[Module[{k = 1}, While[Times @@ Boole@ Map[PrimeQ, {#2^2 + k^2, (#2 + 1)^2 + (k + 1)^2}] < 1, k++]; k] &, 1, Range@ 76] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = my(k=0); while (! (isprime(n^2+k^2) && isprime((n+1)^2+(k+1)^2)), k++); k;  \\ Michel Marcus, Mar 25 2017

A284327 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) - 2)^2 are primes.

Original entry on oeis.org

1, 1, 10, 1, 4, 1, 10, 5, 16, 1, 6, 25, 10, 1, 4, 1, 10, 7, 16, 1, 46, 15, 20, 1, 6, 1, 22, 15, 6, 13, 6, 5, 190, 11, 18, 1, 30, 15, 46, 1, 46, 25, 10, 21, 16, 21, 10, 37, 6, 19, 16, 5, 12, 1, 6, 1, 52, 5, 26, 31, 26, 45, 40, 11, 4, 1, 20, 7, 196, 19, 16

Offset: 1

Lars-Erik Svahn, Mar 25 2017

nonn

n + i*a(n) and n + i*(a(n) - 2) are Gaussian twin primes.

If n^2 + 1 is a prime then a(n) = 1 else a(n) = A284211(n) + 2.

			a(1) = 1 since 1^2 + 1^2 = 2 and 1^2 + (1 - 2)^2 = 2 are primes.

Lars-Erik Svahn, Table of n, a(n) for n = 1..99
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.
Eric Weisstein's World of Mathematics, Gaussian prime
Index entries for Gaussian integers and primes

Cf. A005574, A069003, A284211, A284346.

Rest@ FoldList[Module[{k = 1}, While[Times @@ Boole@ Map[PrimeQ, {#2^2 + k^2, #2^2 + (k - 2)^2}] < 1, k++]; k] &, 1, Range@ 71] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = k=0; while (! (isprime(n^2+k^2) && isprime(n^2+(k-2)^2)), k++); k; \\ Michel Marcus, Mar 25 2017

from sympy import isprime
def a(n):
    k=0
    while True:
        if isprime(n**2 + k**2) and isprime(n**2 + (k - 2)**2): return k
        else: k+=1
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 31 2017

a(n) = 1 for n in A005574. - Michel Marcus, Mar 31 2017

cp's OEIS Frontend

A284346 a(n) is the least positive integer such that n^2 + a(n)^2 and (n + 1)^2 + (a(n) + 1)^2 are primes.

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