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.

A229747 Largest prime factor of 4^(2*n+1)+1.

Original entry on oeis.org

5, 13, 41, 113, 109, 2113, 1613, 1321, 26317, 525313, 14449, 30269, 268501, 279073, 536903681, 384773, 4327489, 47392381, 231769777, 21841, 43249589, 1759217765581, 29247661, 140737471578113, 4981857697937, 1326700741, 1801439824104653, 3630105520141
Offset: 0

Views

Author

Colin Barker, Sep 28 2013

Keywords

Comments

4^(2*n+1)+1 = 2^(2*(2*n+1))+1 = (2^(2*n+1)-2^(n+1)+1) * (2^(2*n+1)+2^(n+1)+1).
For all n, the smallest prime factor of 4^(2*n+1)+1 is 5.
Therefore, the present sequence also gives the largest prime factor of (4^(2*n+1)+1)/5 = A299960(n), for all n > 0. See A299959 for the smallest prime factor of this. - M. F. Hasler, Feb 27 2018

Examples

			For n=7, 4^(2*n+1)+1 = 1073741825 = 5*5*13*41*61*1321. So a(7)=1321.

Links

Crossrefs

Cf. A207262. Bisection of A274903.
Cf. A299960, A299959, A052539.

Programs

  • Mathematica
    Table[FactorInteger[4^(2n+1)+1][[-1,1]],{n,0,30}] (* Harvey P. Dale, Mar 10 2018 *)
  • PARI
    a(n) = {
  f=factor(2^(2*n+1)-2^(n+1)+1);
  g=factor(2^(2*n+1)+2^(n+1)+1);
  max(f[matsize(f)[1],1], g[matsize(g)[1],1])
}

Formula

a(n) = A006530(A052539(2n+1)) = A006530(A207262(n+1)), and for n > 1, a(n) = A006530(A299960(n)) = A006530(A052539(2n+1)/5). \\ M. F. Hasler, Feb 27 2018
a(n) = max(A229767(n), A229768(n)), for n >= 1. - Daniel Suteu, Jun 08 2022

A229768 Largest prime factor of 2^(2*n+1)+2^(n+1)+1.

Original entry on oeis.org

13, 41, 29, 109, 2113, 157, 1321, 26317, 525313, 1429, 1657, 268501, 279073, 536903681, 49477, 4327489, 7416361, 231769777, 21841, 43249589, 500177, 29247661, 7484047069, 19707683773, 1326700741, 586477649, 3630105520141, 275415303169, 104399276341
Offset: 1

Views

Author

Colin Barker, Sep 29 2013

Keywords

Comments

2^(2*n+1)+2^(n+1)+1 is a factor of 4^(2*n+1)+1.

Examples

			For n=10, 2^(2*n+1)+2^(n+1)+1 = 2099201 = 13*113*1429, so a(10)=1429.

Links

Crossrefs

Cf. A085601, A207262, A229747, A229767.

Programs

  • Mathematica
    Table[FactorInteger[2^(2n+1)+2^(n+1)+1][[-1,1]],{n,30}] (* Harvey P. Dale, Nov 03 2017 *)
  • PARI
    a(n) = {f=factor(2^(2*n+1)+2^(n+1)+1); f[matsize(f)[1],1]}

A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

10, 14, 18, 22, 26, 30, 42, 54, 58, 66, 70, 86, 94, 98, 106, 110, 126, 130, 138, 146, 158, 174, 186, 210, 222, 226, 258, 302, 334, 434, 462, 478, 482, 522, 566, 602, 638, 706, 734, 750, 770, 782, 914, 1062, 1086, 1114, 1126, 1226, 1266, 1358, 1382, 1434, 1742, 1926
Offset: 1

Views

Author

Eric Chen, Jan 18 2015

Keywords

Comments

All terms are congruent to 2 modulo 4.
Phi_n(x) is the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nL(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, this is Phi_{2n}(2).
Let L(n) = the Aurifeuillian L-part of 2^n+1, L(n) = 2^(n/2) - 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let L*(n) = GCD(L(n), J*(n)).
This sequence lists all n such that L*(n) is prime.

Examples

			14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
34 is not in this sequence because the left Aurifeuillian primitive part of 2^34+1 is 130561, which equals 137 * 953 and is not prime.

Links

Crossrefs

Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

Programs

  • Mathematica
    Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) - 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]
  • PARI
    isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015
Showing 1-3 of 3 results.

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