A220984 -id:A220984 - cp's OEIS frontend

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

113, 34925927, 4651514210561, 556919483179733591, 65684998500756890925713, 7730533744900130305342957127, 909535949164303794596648514307361, 107006774488854204226839526889653524791, 12589253114717671385404089651370543317211313

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220984.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (137257, -2354918349, 5670354183893, -1944931485075299, 95029449572634843, -651636050170246351, 558545864083284007).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1, {n, 0, 20}]

a(n) = 7^(6n+3) - 7^(5n+3) + 3 * 7^(4n+2) - 7^(3n+2) + 3 * 7^(2n+1) - 7^(n+1) + 1.
Aurifeuillian factorization: 7^(14n+7) + 1 = (7^(2n+1) + 1) * a(n) * A220984(n).
G.f.: -(184010736563880737*x^6 +268740854387875086*x^5 +14564007567924591*x^4 +73553506117028*x^3 +123792021759*x^2 +19415886*x +113) / ((x -1)*(7*x -1)*(49*x -1)*(343*x -1)*(2401*x -1)*(16807*x -1)*(117649*x -1)). [Colin Barker, Jan 04 2013]

A230376 The left Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

Original entry on oeis.org

11, 1803647, 2699538733, 30778903, 112663560435723374699, 554945667652531, 6243610407478181159725577611, 67643278270835231300426724641533, 253382315888712050791030544452181354268272663, 14710826638296122001733445931451

Offset: 1

Colin Barker, Oct 17 2013

nonn

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

Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Apr 13 2022

			1803647 is in the sequence because it is an Aurifeuillian factor of 13^13-1.

Richard P. Brent, On computing factors of cyclotomic polynomials, arXiv:1004.5466 [math.NT], 2010.
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Wikipedia, Léon-François-Antoine Aurifeuille.
Wikipedia, Aurifeuillean factorization.

Cf. A005117, A220983, A220984, A230375, A230377, A230378, A230379.

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

Colin Barker, Oct 17 2013

nonn

The values of k are given by A230375.

Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - Bernard Schott, Apr 25 2022

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

Richard P. Brent, On computing factors of cyclotomic polynomials, arXiv:1004.5466 [math.NT], 2010.
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Wikipedia, Léon-François-Antoine Aurifeuille.
Wikipedia, Aurifeuillean factorization.

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

Colin Barker, Oct 17 2013

nonn

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

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

Richard P. Brent, On computing factors of cyclotomic polynomials, arXiv:1004.5466 [math.NT], 2010.

Cf. A005117, A220983, A220984, A230375-A230377, A230376.

A220986 The right Aurifeuillian factor of 10^(20n + 10) + 1.

Original entry on oeis.org

27961, 1105207205101, 101005020070200501001, 10010005002000700200050010001, 1000100005000200007000200005000100001, 100001000005000020000070000200000500001000001, 10000010000005000002000000700000200000050000010000001

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220985.

Wikipedia, Cunningham Project

Cf. A092440, A085601, A220978, A198410, A220979, A220980, A220981, A220982

Cf. A220983, A220984, A220985, A220987, A220988, A220989, A220990

a[n_] := 10^(8n + 4) + 10^(7n + 4) + 5 * 10^(6n + 3) + 2 * 10^(5n + 3) + 7 * 10^(4n + 2) + 2 * 10^(3n + 2) + 5 * 10^(2n + 1) + 10^(n + 1) + 1

a(n) = 10^(8n + 4) + 10^(7n + 4) + 5 * 10^(6n + 3) + 2 * 10^(5n + 3) + 7 * 10^(4n + 2) + 2 * 10^(3n + 2) + 5 * 10^(2n + 1) + 10^(n + 1) + 1

Aurifeuillian factorization: 10^(20n + 10) + 1 = (10^(4n + 2) + 1) * A220985(n) * a(n)

cp's OEIS Frontend

A220983 The left Aurifeuillian factor of 7^(14n+7) + 1.

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A230376 The left Aurifeuillian factor of k^k - 1 for k congruent to 1 (mod 4) and squarefree.

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

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Author

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Comments

Examples

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

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A220986 The right Aurifeuillian factor of 10^(20n + 10) + 1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula