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.

A352711 The left Aurifeuillian factor of p^p - 1 for primes p congruent to 1 (mod 4).

Original entry on oeis.org

11, 1803647, 2699538733, 112663560435723374699, 6243610407478181159725577611, 67643278270835231300426724641533, 253382315888712050791030544452181354268272663, 133904013361225746608283522164245432446284642589451147, 4429523820749528526448423858097183945539957285504166342434080091097
Offset: 1

Views

Author

Patrick A. Thomas, Mar 30 2022

Keywords

Comments

For prime factorizations of p^p - 1 see A125135.
Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Nov 04 2022

Examples

			112663560435723374699 is the smaller Aurifeuillian factor of 29^29-1, and 29 is the 4th term of A002144, so a(4) = 112663560435723374699.

Links

Crossrefs

Cf. A002144, A125135, A230376, A352732, A352400, A352401.

Formula

If R is (p^p-1)/(p-1), where p == 1 (mod 4) and p > 5, then an approximation of the left Aurifeuillian factor of R is (1/e) * sqrt(R/(1+z)), where z =
2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151 or 169 (mod 210),
2/(3p) + 28/(45p^2) + 86/(2835p^3) if p=19,31,61,139,181 or 199 (mod 210),
2/(3p) - 8/(45p^2) + 194/(2835p^3) if p=37,43,67,127,163 or 193 (mod 210),
2/(3p) - 8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157 or 187 (mod 210),
-2/(3p) - 8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),
-2/(3p) - 8/(45p^2) - 194/(2835p^3) if p=17,47,83,143,167 or 173 (mod 210),
-2/(3p) + 28/(45p^2) - 86/(2835p^3) if p=11,29,71,149,179 or 191 (mod 210),
-2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131 or 209 (mod 210).

A230377 The left Aurifeuillian factor of k^k + 1 for k congruent to 0, 2 or 3 (mod 4) and squarefree.

Original entry on oeis.org

1, 1, 13, 113, 3541, 58367, 2826601, 19231, 113631466919, 9617835527609, 348275601426959, 35522826680397941, 241498479121, 8403855868042458448127, 1161044975606998832441701, 1272844589592126671, 10128165505710094110937686497, 4612290807753604561
Offset: 1

Views

Author

Colin Barker, Oct 17 2013

Keywords

Comments

The values of k are given by A230375.
Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Apr 25 2022

Examples

			58367 is in the sequence because it is an Aurifeuillian factor of 11^11+1.

Links

Crossrefs

Cf. A230375, A220983, A220984, A230375, A230376, A230378, A230379.

A230378 The right Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

Original entry on oeis.org

71, 13993643, 19152352117, 227633407, 813955076015309926319, 4098986195943739, 46959719470144429555105032871, 491873569944394295636860313807677, 1848593595048531176470116001230356265643249547, 108685909290746311448943506365699
Offset: 1

Views

Author

Colin Barker, Oct 17 2013

Keywords

Comments

The values of k are given by A005117, except for the leading 1.

Examples

			13993643 is in the sequence because it is an Aurifeuillian factor of 13^13-1.

Links

Crossrefs

Cf. A005117, A220983, A220984, A230375-A230377, A230376.
Showing 1-3 of 3 results.

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

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