A263466 -id:A263466 - cp's OEIS frontend

A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

9, 11, 19, 21, 23, 25, 29, 31, 39, 41, 43, 49, 51, 53, 55, 59, 61, 63, 69, 71, 75, 77, 79, 81, 83, 89, 91, 93, 99, 101, 105, 107, 109, 111, 113, 119, 121, 123, 127, 129, 131, 133, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161, 165, 169, 171, 173, 175, 179, 181, 185, 187, 189, 191, 195, 197, 199

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

nonn

Conjecture: All terms are odd. An equivalent conjecture in A263977 is that if k > 0 is even, then k^2 + p^2 is prime for some prime p. We have checked this for all k <= 12*10^7.
An odd number k is a term if and only if k^2 + 2^2 is composite, since k^2 + p^2 is even and > 2 (hence composite) for all odd primes p. Thus if the Conjecture is true, then the sequence is simply odd k such that k^2 + 4 is composite. The sequence of odd k, such that k^2 + 4 is prime, is A007591.
The complementary sequence is A263977. Given k in it, the smallest prime p, such that k^2 + p^2 is prime, is in A263726. These numbers k^2 + p^2 form A185086, the Fouvry-Iwaniec primes.

			9^2 + 2^2 = 85 = 5*17, 11^2 + 2^2 = 125 = 5^3, and 23^2 + 2^2 = 533 = 13*41 are composite, so 9, 11, and 23 are members.
1^2 + 2^2 = 5 and 2^2 + 3^3 = 13 are prime, so 1, 2, and 3 are not members.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, arXiv:math/0703284 [math.NT], 2007-2008; Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A002313, A045637, A062324, A185086, A007591, A240130, A240131, A263466, A263726, A263977.

A263726 Least prime p such that p^2 + A263977(n)^2 is prime.

Original entry on oeis.org

2, 3, 2, 5, 2, 5, 2, 3, 3, 7, 2, 11, 2, 5, 2, 5, 3, 5, 5, 5, 2, 5, 11, 3, 2, 5, 2, 5, 2, 3, 3, 5, 19, 2, 5, 2, 13, 7, 3, 11, 11, 2, 3, 13, 3, 11, 2, 29, 2, 5, 3, 5, 2, 5, 5, 7, 7, 3, 11, 2, 11, 2, 3, 11, 7, 5, 2, 5, 2, 3, 3, 5, 2, 11, 5, 5, 3, 3, 59, 2, 11, 2, 3, 7, 13, 5, 2, 5, 7

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 30 2015

nonn

The least k, such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, is in A263466.

			A263977(1) = 1, and 2 and 2^2 + 1^2 = 5 are prime, so a(1) = 2.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A240130, A240131, A263466, A263722, A263977.

f[n_] := Block[{p = 2}, While[! PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@ p]; If[p > 1500, 0, p]]; lst = {}; k = 1; While[k < 130, If[f@ k > 0, AppendTo[lst, f@ k]]; k++]; lst

A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 97, 98, 100, 102, 103, 104, 106, 108, 110, 112, 114, 115, 116, 117, 118, 120, 122, 124, 125, 126

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 30 2015

nonn

The smallest such prime p is in A263726.
Complement of A263722.
An odd number k is a member if and only if k^2 + 4 is prime; see A007591.
Conjecture: Every even number k is a member. (This is equivalent to the Conjecture in A263722.) We have checked this for all k <= 12*10^7.

			1^2 + 2^2 = 5, and 2 and 5 are prime, so a(1) = 1.
9^2 + p^2 is composite for all primes p, so 9 is not a member.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A007591, A240130, A240131, A263466, A263722, A263726.

fQ[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@ p]; If[p > 1500, 0, p]]; lst = {}; k = 1; While[k < 130, If[fQ@ k > 0, AppendTo[lst, k]]; k++]; lst

A263978 Least prime p such that n^2 + p^2 is prime, or 0 if none.

Original entry on oeis.org

2, 3, 2, 5, 2, 5, 2, 3, 0, 3, 0, 7, 2, 11, 2, 5, 2, 5, 0, 3, 0, 5, 0, 5, 0, 5, 2, 5, 0, 11, 0, 3, 2, 5, 2, 5, 2, 3, 0, 3, 0, 5, 0, 19, 2, 5, 2, 13, 0, 7, 0, 3, 0, 11, 0, 11, 2, 3, 0, 13, 0, 3, 0, 11, 2, 29, 2, 5, 0, 3, 0, 5, 2, 5, 0, 5, 0, 7, 0, 7, 0, 3, 0, 11, 2, 11, 2, 3, 0, 11, 0, 7, 0, 5, 2, 5, 2, 3, 0, 3

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Nov 06 2015

nonn

When n is odd, n^2 + p^2 is composite for all odd primes p, so a(n) = 2 or 0 according as n^2 + 2^2 is prime or not.
The locations of the zeros are in A263722.
The location of the first occurrence of prime(n) is A263466(n).

			a(1) = 2 since 1^2 + 2^2 = 5 is prime.
a(2) = 3 since 2^2 + 2^2 = 8 is not prime but 2^2 + 3^2 = 13 is prime.
a(9) = 0 since 9^2 + 2^2 = 85 is not prime.

Cf. A240130, A240131, A263466, A263721, A263722, A263726, A263977.

f[n_] := If[OddQ[n] && ! PrimeQ[n^2 + 4], 0,
  Block[{p = 2}, While[! PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@p];
   p]]; Array[f, 100]

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A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

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A263726 Least prime p such that p^2 + A263977(n)^2 is prime.

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A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p.

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A263978 Least prime p such that n^2 + p^2 is prime, or 0 if none.

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