A263722 -id:A263722 - cp's OEIS frontend

A263466 Least k such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, or 0 if none.

1, 2, 4, 12, 14, 48, 178, 44, 152, 66, 224, 272, 496, 322, 408, 2068, 114, 354, 592, 584, 3192, 406, 2708, 774, 2658, 394, 4102, 2432, 3346, 2562, 8722, 4424, 9562, 2986, 6856, 1714, 21318, 5858, 7568, 16272, 7576, 4864, 6244, 29262, 29992, 9996, 10406, 58348, 16872, 11384, 12738, 22126, 9946, 24214, 81682, 46082, 74616, 88016, 6788, 30856, 21542, 38672, 131492, 62874, 75358, 95262, 39554, 83552, 65022, 73664

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Nov 02 2015

nonn

The data support the conjecture in A263977 that if k > 0 is even, then k^2 + p^2 is prime for some prime p.

a(n) is the location of the first occurrence of prime(n) in A263978.

			The primes p < prime(3) = 5 are p = 2 and 3. As 1^2 + 2^2 = 5, 2^2 + 3^2 = 13, and 3^2 + 2^2 = 13 are prime, a(3) >= 4. But 4^2 + 2^2 = 20 and 4^2 + 3^2 = 25 are not prime, while 4^2 + 5^2 = 41 is prime, so a(3) = 4.

Jonathan Sondow and Robert G. Wilson v, Table of n, a(n) for n = 1..260
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, A263721, A263722, A263726, A263977, A263978.

f[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2], p = NextPrime@ p]; p]; t = 0*Range@ 300; t[[1]] = 1; k = 2; While[k < 50000001, p = f@ k; If[ t[[PrimePi@ p]] == 0, t[[PrimePi@ p]] = k; Print[{PrimePi@ p, p, k}]]; k += 2]; t

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

A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

Original entry on oeis.org

2, 3, 5, 5, 7, 5, 3, 5, 3, 7, 11, 7, 11, 13, 7, 2, 13, 13, 5, 17, 13, 11, 5, 17, 7, 17, 19, 3, 17, 7, 19, 5, 11, 19, 13, 23, 5, 17, 19, 13, 23, 5, 2, 19, 17, 11, 5, 23, 29, 29, 23, 19, 29, 13, 31, 31, 23, 11, 3, 5, 31, 13, 2, 29, 5, 13, 31, 2, 11, 5, 31, 37, 23, 37, 3, 7, 23, 3, 13, 31, 19, 37, 41, 11

Offset: 1

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

nonn

The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.

The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.

			A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.

p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]

do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ Charles R Greathouse IV, Aug 21 2017

a(n)^2 = A185086(n) - k^2 for some integer k > 0.

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]

cp's OEIS Frontend

A263466 Least k such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, or 0 if none.

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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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A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

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

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