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A125611 a(n) is the smallest prime p such that 7^n divides p^6 - 1.

%I A125611 #31 May 02 2024 17:33:41
%S A125611 2,19,19,3449,32261,152617,3294173,3376853,135967277,135967277,
%T A125611 7909306973,92233439147,115385868869,1356446145697,56020344873707,
%U A125611 56020344873707,930522055948829,9116268492336169,10744682090246617
%N A125611 a(n) is the smallest prime p such that 7^n divides p^6 - 1.
%C A125611 a(n) is the smallest 6th root of unity (mod 7^n) that is prime. - _Robert Israel_, Jan 14 2024
%H A125611 Robert Israel, <a href="/A125611/b125611.txt">Table of n, a(n) for n = 1..1000</a>
%H A125611 Wilfrid Keller and Jörg Richstein, <a href="https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients that are divisible by p</a>, 2014. [Wayback Machine link]
%p A125611 f:= proc(n) local R, r, i;
%p A125611   R:= sort(map(rhs@op, [msolve(x^6=1, 7^n)]));
%p A125611   for i from 0 do
%p A125611     for r in R do
%p A125611       if isprime(7^n * i + r) then return 7^n * i + r fi
%p A125611   od od;
%p A125611 end proc:
%p A125611 map(f, [$1..30]); # _Robert Israel_, Jan 14 2024
%o A125611 (PARI) \\ See A125609
%o A125611 (Python)
%o A125611 from itertools import count
%o A125611 from sympy import nthroot_mod, isprime
%o A125611 def A125611(n):
%o A125611     m = 7**n
%o A125611     r = sorted(nthroot_mod(1,6,m,all_roots=True))
%o A125611     for i in count(0,m):
%o A125611         for p in r:
%o A125611             if isprime(i+p): return i+p # _Chai Wah Wu_, May 02 2024
%Y A125611 Cf. A125609, A125610, A125612, A125632, A125633, A125634, A125635.
%K A125611 nonn
%O A125611 1,1
%A A125611 _Alexander Adamchuk_, Nov 28 2006
%E A125611 More terms from _Ryan Propper_, Jan 03 2007
%E A125611 More terms from _Martin Fuller_, Jan 11 2007