Lars-Erik Svahn - cp's OEIS frontend

A179295 a(n) is the least prime number such that prime(n)+a(n)+1 is a prime or -1 if no such prime number exists.

2, 3, 5, 3, 5, 3, 5, 3, 5, 7, 5, 3, 5, 3, 5, 5, 7, 5, 3, 7, 5, 3, 5, 7, 3, 5, 3, 5, 3, 13, 3, 5, 11, 11, 7, 5, 5, 3, 5, 5, 11, 11, 5, 3, 13, 11, 11, 3, 5, 3, 5, 11, 29, 5, 5, 5, 7, 5, 3, 11, 23, 13, 3, 5, 3, 13, 5, 11, 5, 3, 5, 7, 5, 5, 3, 5, 7, 3, 7, 11, 11

Offset: 1

Lars-Erik Svahn, Jun 21 2017

nonn

If Maillet's conjecture is true, then a(n) != -1 for all n. - Chai Wah Wu, Aug 01 2017

			a(1) = 2, since prime(1) + 2 + 1 = 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Wen Huang and XiaoSheng Wu, On the set of the difference of primes, Proc. Amer. Math. Soc. 145 (2017), 3787-3793.
E. Maillet, Réponse, L’intermédiaire des math. 12 (1905), p. 108.

Table[Block[{p=2}, While[!PrimeQ[Prime[n] + p + 1], p=NextPrime[p]]; p],{n, 100}] (* Indranil Ghosh, Jun 30 2017 *)

a(n) = my(pn=prime(n), p=2); while(! isprime(pn+p+1), p = nextprime(p+1)); p; \\ Michel Marcus, Jun 30 2017

from sympy import prime, isprime, nextprime
def a(n):
    p=2
    while not isprime(prime(n) + p + 1): p=nextprime(p)
    return p
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 30 2017

Definition clarified by Chai Wah Wu, Aug 01 2017

More terms from Chai Wah Wu, Aug 02 2017

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.

Original entry on oeis.org

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

A284376 a(n) is the least nonnegative integer such that n + i*a(n) is a Gaussian prime.

Original entry on oeis.org

3, 1, 1, 0, 1, 2, 1, 0, 3, 4, 1, 0, 7, 2, 1, 2, 1, 2, 5, 0, 1, 4, 5, 0, 1, 4, 1, 2, 5, 4, 11, 0, 3, 2, 5, 2, 1, 2, 3, 10, 1, 4, 5, 0, 9, 2, 5, 0, 13, 4, 7, 4, 3, 10, 1, 4, 1, 2, 3, 0, 13, 10, 3, 32, 9, 2, 1, 0, 5, 10, 3, 0, 5, 2, 1, 4, 5, 10, 7, 0, 7, 4, 3, 0, 1, 2, 9, 2, 3, 4, 1, 4, 7, 8, 1, 2, 5, 2, 3, 4, 3

Offset: 0

Lars-Erik Svahn, Mar 25 2017

nonn

Lars-Erik Svahn, Table of n, a(n) for n = 0..10000
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.
Wikipedia, Gaussian prime
Index entries for Gaussian integers and primes

Cf. A002145, A005574, A069003.

f:= proc(n) local k;
  for k from 0 do if GaussInt:-GIprime(n+I*k) then return k fi od
end proc:
map(f, [$0..100]); # Robert Israel, Apr 07 2017

Table[k = 0; While[! PrimeQ[n + I k, GaussianIntegers -> True], k++]; k, {n, 0, 100}] (* Michael De Vlieger, Mar 29 2017 *)

From Michel Marcus, Mar 30 2017: (Start)
a(n) = 0 for n in A002145.
a(n) = 1 for n in A005574.
(End)
a(n) = A069003(n) if n is not in A002145. - Robert Israel, Apr 07 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

A284211 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

2, 1, 8, 9, 2, 29, 8, 3, 14, 1, 4, 23, 8, 9, 2, 29, 8, 5, 14, 1, 44, 13, 18, 59, 4, 9, 20, 13, 4, 11, 4, 3, 188, 9, 16, 149, 28, 13, 44, 1, 44, 23, 8, 19, 14, 19, 8, 35, 4, 17, 14, 3, 10, 59, 4, 9, 50, 3, 24, 29, 24, 43, 38, 9, 2, 89, 18, 5, 194, 17, 14, 5

Offset: 1

Lars-Erik Svahn, Mar 23 2017

nonn

z = n + i*a(n) and z' = n + i*(a(n) + 2) are two Gaussian primes such that |z - z'| = 2, corresponding to twin primes.

			a(1)=2: 1^2 + 1^2 = 2 is a prime but 1 + (1 + 2)^2 = 10 is not, while 1^2 + 2^2 = 5 and 1^2 + (2+2)^2 = 17 are both primes.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 99 terms from Lars-Erik Svahn)
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, J. of Number Theory 171 (2017), 449-473.

Cf. A069003.

f:= proc(n) local k,pp,p;
    pp:= false;
    for k from (n+1) mod 2 by 2 do
      p:= isprime(n^2 + k^2);
      if p and pp then return k-2 fi;
      pp:= p;
    od;
end proc:
map(f, [$1..100]); # Robert Israel, Mar 30 2017

a[n_] := Block[{k = Mod[n, 2] + 1}, While[! PrimeQ[n^2 + k^2] || ! PrimeQ[n^2 + (k + 2)^2], k += 2]; k]; Array[a, 72] (* Giovanni Resta, Mar 23 2017 *)
lpi[n_]:=Module[{k=1},While[!AllTrue[n^2+{k^2,(k+2)^2},PrimeQ],k++];k]; Array[lpi,80] (* Harvey P. Dale, Aug 15 2024 *)

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

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User: Lars-Erik Svahn

A179295 a(n) is the least prime number such that prime(n)+a(n)+1 is a prime or -1 if no such prime number exists.

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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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A284376 a(n) is the least nonnegative integer such that n + i*a(n) is a Gaussian prime.

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

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A284211 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) + 2)^2 are primes.

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Maple

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