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 1-3 of 3 results.

A304903 Least prime p such that 2*n^2 - p is prime.

Original entry on oeis.org

3, 5, 3, 3, 5, 19, 19, 5, 3, 3, 5, 7, 3, 7, 3, 7, 5, 3, 3, 5, 31, 7, 23, 13, 31, 5, 19, 13, 11, 43, 19, 17, 3, 3, 13, 7, 31, 5, 13, 3, 11, 7, 19, 23, 3, 61, 5, 3, 7, 5, 61, 37, 5, 3, 3, 7, 19, 3, 7, 31, 7, 5, 13, 3, 5
Offset: 2

Views

Author

Hugo Pfoertner, May 20 2018

Keywords

Comments

Each square > 1 can be written as the average of 2 primes p1 < p2. a(n) gives the least prime p1 such that n^2 = (p1 + p2) / 2. The corresponding p2 is provided in A304904.

Examples

			a(5) = 3 because 2*5^2 - 3 = 47 is prime,
a(7) = 19 because 2*7^2 - 19 = 71 is prime, whereas all of 98 - 3 = 95, 98 - 5 = 93, 98 - 7 = 91, 98 - 11 = 87, 98 - 13 = 85 and 98 - 17 = 81 are composite.

Crossrefs

Cf. A304874, A304875, A304904, A304905.

Programs

  • Mathematica
    f[n_] := Block[{p = 2}, While[ !PrimeQ[2 n^2 - p], p = NextPrime@ p]; p]; Array[f, 65, 2] (* Robert G. Wilson v, May 20 2018 *)
  • PARI
    a(n) = forprime(p=3, , if(ispseudoprime(2*n^2-p), return(p))) \\ Felix Fröhlich, May 20 2018

Formula

a(n) = n^2 - A304905(n) = A304904(n) - 2*A304905(n).

A304905 Greatest difference d such that both n^2 - d and n^2 + d are primes.

Original entry on oeis.org

1, 4, 13, 22, 31, 30, 45, 76, 97, 118, 139, 162, 193, 218, 253, 282, 319, 358, 397, 436, 453, 522, 553, 612, 645, 724, 765, 828, 889, 918, 1005, 1072, 1153, 1222, 1283, 1362, 1413, 1516, 1587, 1678, 1753, 1842, 1917
Offset: 2

Views

Author

Hugo Pfoertner, May 20 2018

Keywords

Examples

			a(2) = 1 because 2^2 - 1 = 3 and 2^2 + 1 = 5 are primes.
a(7) = 30 because 7^2 - 30 = 19 and 7^2 + 30 = 79 is the pair with maximum difference. All greater differences lead to at least one composite, i.e., 49 + 32 = 81, 49 - 34 = 15, 49 + 36 = 85, 49 + 38 = 87, 49 - 40 = 9, 49 + 42 = 91 = 7*13, 49 + 44 = 93 = 3*31, 49 + 46 = 95, and 49 - 48 = 1 is not a prime.

Crossrefs

Cf. A172989, A304903, A304904.

Programs

  • PARI
    a304903(n) = forprime(p=3, , if(ispseudoprime(2*n^2-p), return(p)))
a(n) = n^2 - a304903(n) \\ Felix Fröhlich, May 20 2018

Formula

a(n) = (A304904(n) - A304903(n))/2 = n^2 - A304903(n) = A304904(n) - n^2.

A305125 Number of ways to express n^2 as average of two primes p1 < p2.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 3, 3, 10, 8, 8, 17, 9, 11, 27, 11, 12, 27, 14, 21, 39, 17, 19, 36, 28, 22, 48, 25, 24, 75, 30, 25, 68, 35, 56, 68, 37, 40, 93, 54, 43, 103, 42, 52, 125, 51, 49, 117, 64, 76, 130, 63, 56, 135, 99, 78, 151, 76, 73, 198
Offset: 1

Views

Author

Hugo Pfoertner, May 26 2018

Keywords

Links

Crossrefs

Cf. A002375, A172989, A304874, A304875, A304903, A304904, A304905, A305126.

Programs

  • Mathematica
    f[n_]:=Length[Select[2 n - Prime[Range[2, PrimePi[n]]], PrimeQ]]; Table[f[n^2], {n, 60}] (* Vincenzo Librandi, May 26 2018 *)
  • PARI
    for (n=1,60,k=0;n2=2*n*n;forprime(p=3,n2/2,if(isprime(n2-p),k++));print1(k,", "))

Formula

a(n) = A002375(n^2).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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