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).

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

Offset

1,2

Comments

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

Links

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

Calculators, Aurifeuillian LMs

Wikipedia, Aurifeuillean factorization.

Formula

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).

Example

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

Crossrefs

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

Sequence in context: A152496 A176825 A187703 * A329105 A304324 A180052

Adjacent sequences: A352397 A352398 A352399 * A352401 A352402 A352403

Keyword

nonn

Author

Patrick A. Thomas, Jun 08 2022

Status

approved