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

1,2

Comments

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.

Links

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.

Example

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.

Mathematica

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

Crossrefs

Cf. A240130, A240131, A263721, A263722, A263726, A263977, A263978.

Sequence in context: A107295 A039564 A297064 * A106135 A067268 A285378

Adjacent sequences: A263463 A263464 A263465 * A263467 A263468 A263469

Keyword

nonn

Author

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

Status

approved