cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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

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.

Original entry on oeis.org

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

Views

Author

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

Keywords

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • 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

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

Links

  • 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.

Crossrefs

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

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

Views

Author

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

Keywords

Comments

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).

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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]
Showing 1-4 of 4 results.

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