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A125612 a(n) is the smallest prime p such that 11^n divides p^10 - 1.

%I A125612 #29 May 02 2024 17:33:35
%S A125612 2,3,2663,45989,275393,2120879,28723679,174625993,4715895383,
%T A125612 24262286441,1194631280321,3143820659087,138090848575723,
%U A125612 488581592070877,6266190914259137,367597838908577287,10866698414795559631,19697814061539637951,19697814061539637951,3824465353837845574717,14852046860008834240157
%N A125612 a(n) is the smallest prime p such that 11^n divides p^10 - 1.
%C A125612 a(n) is the smallest 10th root of unity (mod 11^n) that is prime. - _Robert Israel_, Jan 14 2024
%H A125612 Robert Israel, <a href="/A125612/b125612.txt">Table of n, a(n) for n = 1..954</a>
%H A125612 W. Keller and J. Richstein, <a href="http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients that are divisible by p</a>.
%p A125612 f:= proc(n) local R,r,i;
%p A125612   R:= sort(map(rhs@op, [msolve(x^10=1, 11^n)]));
%p A125612   for i from 0 do
%p A125612     for r in R do
%p A125612       if isprime(11^n * i + r) then return 11^n * i + r fi
%p A125612   od od;
%p A125612 end proc:
%p A125612 map(f, [$1..20]); # _Robert Israel_, Jan 14 2024
%t A125612 spp[n_]:=Module[{p=2,c=11^n},While[PowerMod[p,10,c]!=1,p=NextPrime[p]];p]; Array[spp,16] (* _Harvey P. Dale_, Aug 08 2019 *)
%o A125612 (PARI) \\ See A125609
%o A125612 (Python)
%o A125612 from itertools import count
%o A125612 from sympy import nthroot_mod, isprime
%o A125612 def A125612(n):
%o A125612     m = 11**n
%o A125612     r = sorted(nthroot_mod(1,10,m,all_roots=True))
%o A125612     for i in count(0,m):
%o A125612         for p in r:
%o A125612             if isprime(i+p): return i+p # _Chai Wah Wu_, May 02 2024
%Y A125612 Cf. A125609 = Smallest prime p such that 3^n divides p^2 - 1. Cf. A125610 = Smallest prime p such that 5^n divides p^4 - 1. Cf. A125611 = Smallest prime p such that 7^n divides p^6 - 1. Cf. A125632 = Smallest prime p such that 13^n divides p^12 - 1. Cf. A125633 = Smallest prime p such that 17^n divides p^16 - 1. Cf. A125634 = Smallest prime p such that 19^n divides p^18 - 1. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.
%K A125612 nonn
%O A125612 1,1
%A A125612 _Alexander Adamchuk_, Nov 28 2006
%E A125612 More terms from _Ryan Propper_, Jan 03 2007
%E A125612 More terms from _Martin Fuller_, Jan 11 2007
%E A125612 More terms from _Robert Israel_, Jan 14 2024