A352732 -id:A352732 - cp's OEIS frontend

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

11, 1803647, 2699538733, 112663560435723374699, 6243610407478181159725577611, 67643278270835231300426724641533, 253382315888712050791030544452181354268272663, 133904013361225746608283522164245432446284642589451147, 4429523820749528526448423858097183945539957285504166342434080091097

Offset: 1

Patrick A. Thomas, Mar 30 2022

nonn

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

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

Patrick A. Thomas, Table of n, a(n) for n = 1..60
Calculators, Aurifeuillian LMs
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Wikipedia, Léon-François-Antoine Aurifeuille.
Wikipedia, Aurifeuillean factorization.

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

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).

A352400 a(n) is the left Aurifeuillian factor of p^p + 1 for A002145(n), where A002145 lists the primes congruent to 3 (mod 4).

Original entry on oeis.org

1, 113, 58367, 113631466919, 348275601426959, 8403855868042458448127, 7248206084007410402911299180581471, 105318477338066161993242388018074119617, 830220061043693789623432394289631761145130727636121

Offset: 1

Patrick A. Thomas, Jun 08 2022

nonn

For prime factorizations of p^p + 1 see A125136.

			105318477338066161993242388018074119617 is the smaller Aurifeuillian factor of 47^47 + 1, and 47 is the 8th term of A002145, so it is a(8).

Patrick A. Thomas, Table of n, a(n) for n = 1..65
Calculators, Aurifeuillian LMs
Wikipedia, Aurifeuillean factorization.

Cf. A002145, A125136, A230377, A352711, A352732, A352401.

If R is (p^p+1)/(p+1), where p == 3 (mod 4) and p > 7, 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).

A352401 The right Aurifeuillian factor of p^p + 1 for primes p congruent to 3 (mod 4).

Original entry on oeis.org

7, 911, 407353, 870542161121, 2498077661567473, 63472256064447557254913, 54382651771205224279713471565249817, 767102704711961850109296220485687497279, 6066304600323886604542912453739225327712511596287519

Offset: 1

Patrick A. Thomas, Jun 08 2022

nonn

For prime factorizations of p^p + 1 see A125136.

			870542161121 is the larger Aurifeuillian factor of 19^19 + 1, and 19 is the 4th term of A002145, so a(4) = 870542161121.

Patrick A. Thomas, Table of n, a(n) for n = 1..65
Calculators, Aurifeuillian LMs

Cf. A002145, A125136, A230379, A352711, A352732, A352400.

A356518 Maximal numerators in approximations to the Aurifeuillian factors of p^p +- 1.

Original entry on oeis.org

2, 28, 1706, 25082, 816634, 157704814

Offset: 1

Patrick A. Thomas, Aug 10 2022

If R=(p^p+-1)/(p+-1) then the left Aurifeuillian factor of R is (1/e)*sqrt(R/(1+z)), where z = Sum_{n>=1} r(n)/p^n and r(n) is a rational number in Q; a(n) is the numerator of r(n). r(7) appears to be about 0.572082, but there is insufficient precision to identify a(7).

			The r(n) are 2/3, 28/45, 1706/2835, 25082/42525, 816634/1403325, ...

Cf. A356519 (denominators), A352711, A352732, A352400, A352401.

A356519 Denominators in approximations to the Aurifeuillian factors of p^p +- 1.

Original entry on oeis.org

3, 45, 2835, 42525, 1403325, 273648375

Offset: 1

Patrick A. Thomas, Aug 10 2022

a(1) = 3                  =  4! / 2^3
a(2) = 3^2 * 5            =  6! / 2^4
a(3) = 3^4 * 5 * 7        =  8! / 2^7 * 3^2
a(4) = 3^5 * 5^2 * 7      = 10! / 2^8 * 3
a(5) = 3^6 * 5^2 * 7 * 11 = 12! / 2^10 * 3

Cf. A356518 (numerators), A352711, A352732, A352400, A352401.

cp's OEIS Frontend

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

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A352400 a(n) is the left Aurifeuillian factor of p^p + 1 for A002145(n), where A002145 lists the primes congruent to 3 (mod 4).

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A352401 The right Aurifeuillian factor of p^p + 1 for primes p congruent to 3 (mod 4).

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A356518 Maximal numerators in approximations to the Aurifeuillian factors of p^p +- 1.

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A356519 Denominators in approximations to the Aurifeuillian factors of p^p +- 1.

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