%I #74 Sep 10 2022 06:26:42

%S 1,113,58367,113631466919,348275601426959,8403855868042458448127,

%T 7248206084007410402911299180581471,

%U 105318477338066161993242388018074119617,830220061043693789623432394289631761145130727636121

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

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

%H Patrick A. Thomas, <a href="/A352400/b352400.txt">Table of n, a(n) for n = 1..65</a>

%H Calculators, <a href="http://myfactorcollection.mooo.com:8090/calculators.html">Aurifeuillian LMs</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Aurifeuillean_factorization">Aurifeuillean factorization</a>.

%F 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 =

%F 2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151 or 169 (mod 210),

%F 2/(3p) + 28/(45p^2) + 86/(2835p^3) if p=19,31,61,139,181 or 199 (mod 210),

%F 2/(3p) - 8/(45p^2) + 194/(2835p^3) if p=37,43,67,127,163 or 193 (mod 210),

%F 2/(3p) - 8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157 or 187 (mod 210),

%F -2/(3p) - 8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),

%F -2/(3p) - 8/(45p^2) - 194/(2835p^3) if p=17,47,83,143,167 or 173 (mod 210),

%F -2/(3p) + 28/(45p^2) - 86/(2835p^3) if p=11,29,71,149,179 or 191 (mod 210),

%F -2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131 or 209 (mod 210).

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

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

%K nonn

%O 1,2

%A _Patrick A. Thomas_, Jun 08 2022