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.

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A302089 Primes of form 5*k^k + 4.

Original entry on oeis.org

139, 15629, 1937102449
Offset: 1

Views

Author

Seiichi Manyama, Apr 01 2018

Keywords

Comments

The next term is too large to include.

Crossrefs

Primes of form b*k^k + b - 1: A216148 (b=2), A301811 (b=3), A301808 (b=4), this sequence (b=5), A302091 (b=6).
Cf. A302088.

Programs

  • PARI
    lista(nn) = forstep(n=1, nn, 2, if(ispseudoprime(p=5*n^n+4), print1(p, ", "))); \\ Altug Alkan, Apr 01 2018

Formula

a(n) = 5*A302088(n)^A302088(n) + 4.

A174711 Composites of the form 2*n^n + 1 = A216147(n).

Original entry on oeis.org

9, 55, 513, 6251, 93313, 1647087, 33554433, 774840979, 20000000001, 570623341223, 605750213184507, 22224013651116033, 875787780761718751, 36893488147419103233, 1654480523772673528355, 3956839311320627178247959
Offset: 1

Views

Author

Michel Lagneau, Mar 27 2010

Keywords

Comments

If p = n+2 is prime, then p divides 2*n^n + 1. Proof: Let p = n+2 prime. Then, according to Fermat's theorem, n^(p-1) == 1 (mod p). Because p-1 = n+1, n^(n+1) == 1 (mod p), and with n = p-2 == -2 (mod p), we obtain successively: n*n^n == 1 (mod p), -2*n^n == 1 (mod p), 2*n^n == -1 (mod p) => p divides 2*n^n + 1.

Examples

			a(2) = 9 = 3^2, a(3) = 55 = 5*11, a(4) = 513 = 3 ^ 3 * 19.

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
  • J. M. De Koninck, A. Mercier, 1001 problemes en theorie classique des nombres, Ellipses 2004, p. 52.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

Links

Crossrefs

Complement of A216148 in A216147. - M. F. Hasler, Sep 02 2012

Programs

  • Maple
    with(numtheory):for n from 0 to 50 do: x:=2*n^n + 1 : if type(x,prime)=false then print (x):else fi:od:
  • Mathematica
    Select[Table[2n^n+1,{n,20}],CompositeQ] (* Harvey P. Dale, Jun 21 2015 *)
Previous Showing 11-12 of 12 results.

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