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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).

%I A352400 #74 Sep 10 2022 06:26:42
%S A352400 1,113,58367,113631466919,348275601426959,8403855868042458448127,
%T A352400 7248206084007410402911299180581471,
%U A352400 105318477338066161993242388018074119617,830220061043693789623432394289631761145130727636121
%N 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).
%C A352400 For prime factorizations of p^p + 1 see A125136.
%H A352400 Patrick A. Thomas, <a href="/A352400/b352400.txt">Table of n, a(n) for n = 1..65</a>
%H A352400 Calculators, <a href="http://myfactorcollection.mooo.com:8090/calculators.html">Aurifeuillian LMs</a>
%H A352400 Wikipedia, <a href="https://en.wikipedia.org/wiki/Aurifeuillean_factorization">Aurifeuillean factorization</a>.
%F A352400 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 A352400    2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151  or 169 (mod 210),
%F A352400    2/(3p) + 28/(45p^2) +   86/(2835p^3) if p=19,31,61,139,181  or 199 (mod 210),
%F A352400    2/(3p) -  8/(45p^2) +  194/(2835p^3) if p=37,43,67,127,163  or 193 (mod 210),
%F A352400    2/(3p) -  8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157  or 187 (mod 210),
%F A352400   -2/(3p) -  8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),
%F A352400   -2/(3p) -  8/(45p^2) -  194/(2835p^3) if p=17,47,83,143,167  or 173 (mod 210),
%F A352400   -2/(3p) + 28/(45p^2) -   86/(2835p^3) if p=11,29,71,149,179  or 191 (mod 210),
%F A352400   -2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131  or 209 (mod 210).
%e A352400 105318477338066161993242388018074119617 is the smaller Aurifeuillian factor of 47^47 + 1, and 47 is the 8th term of A002145, so it is a(8).
%Y A352400 Cf. A002145, A125136, A230377, A352711, A352732, A352401.
%K A352400 nonn
%O A352400 1,2
%A A352400 _Patrick A. Thomas_, Jun 08 2022