OFFSET
1,1
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
LINKS
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.
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).
EXAMPLE
112663560435723374699 is the smaller Aurifeuillian factor of 29^29-1, and 29 is the 4th term of A002144, so a(4) = 112663560435723374699.
CROSSREFS
KEYWORD
nonn
AUTHOR
Patrick A. Thomas, Mar 30 2022
STATUS
approved