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

%I A352711 #57 Nov 04 2022 07:30:21
%S A352711 11,1803647,2699538733,112663560435723374699,
%T A352711 6243610407478181159725577611,67643278270835231300426724641533,
%U A352711 253382315888712050791030544452181354268272663,133904013361225746608283522164245432446284642589451147,4429523820749528526448423858097183945539957285504166342434080091097
%N A352711 The left Aurifeuillian factor of p^p - 1 for primes p congruent to 1 (mod 4).
%C A352711 For prime factorizations of p^p - 1 see A125135.
%C A352711 Named after the French mathematician Léon-François-Antoine Aurifeuille (1822-1882). - _Bernard Schott_, Nov 04 2022
%H A352711 Patrick A. Thomas, <a href="/A352711/b352711.txt">Table of n, a(n) for n = 1..60</a>
%H A352711 Calculators, <a href="http://myfactorcollection.mooo.com:8090/calculators.html">Aurifeuillian LMs</a>
%H A352711 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AurifeuilleanFactorization.html">Aurifeuillean Factorization</a>.
%H A352711 Wikipedia, <a href="https://en.wikipedia.org/wiki/Léon-François-Antoine_Aurifeuille">Léon-François-Antoine Aurifeuille</a>.
%H A352711 Wikipedia, <a href="https://en.wikipedia.org/wiki/Aurifeuillean_factorization">Aurifeuillean factorization</a>.
%F A352711 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 =
%F A352711    2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151  or 169 (mod 210),
%F A352711    2/(3p) + 28/(45p^2) +   86/(2835p^3) if p=19,31,61,139,181  or 199 (mod 210),
%F A352711    2/(3p) -  8/(45p^2) +  194/(2835p^3) if p=37,43,67,127,163  or 193 (mod 210),
%F A352711    2/(3p) -  8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157  or 187 (mod 210),
%F A352711   -2/(3p) -  8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),
%F A352711   -2/(3p) -  8/(45p^2) -  194/(2835p^3) if p=17,47,83,143,167  or 173 (mod 210),
%F A352711   -2/(3p) + 28/(45p^2) -   86/(2835p^3) if p=11,29,71,149,179  or 191 (mod 210),
%F A352711   -2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131  or 209 (mod 210).
%e A352711 112663560435723374699 is the smaller Aurifeuillian factor of 29^29-1, and 29 is the 4th term of A002144, so a(4) = 112663560435723374699.
%Y A352711 Cf. A002144, A125135, A230376, A352732, A352400, A352401.
%K A352711 nonn
%O A352711 1,1
%A A352711 _Patrick A. Thomas_, Mar 30 2022