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Prime divisors of solutions to 10^n == 1 (mod n).
10

%I #41 Aug 18 2024 20:12:19

%S 3,37,163,757,1999,5477,8803,9397,13627,15649,36187,40879,62597,

%T 106277,147853,161839,215893,231643,281683,295759,313471,333667,

%U 338293,478243,490573,607837,647357,743933,988643,1014877,1056241,1168711,1353173,1390757,1487867,1519591,1627523,1835083,1912969,2028119,2029759,2064529

%N Prime divisors of solutions to 10^n == 1 (mod n).

%H Max Alekseyev and Hans Havermann (Max Alekseyev to 501), <a href="/A066364/b066364.txt">Table of n, a(n) for n = 1..2060</a>

%H RĂ¼diger Jehn and Kester Habermann, <a href="https://arxiv.org/abs/2106.05866">Properties of terms of OEIS A342810</a>, arXiv:2106.05866 [math.GM], 2021.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit">Factorizations of 11...11 (Repunit)</a>.

%F A prime p is a term iff all prime divisors of ord_p(10) are terms, where ord_p(10) is the order of 10 modulo p. - _Max Alekseyev_, Nov 16 2005

%e 10^27-1 = 3^5*37*757*333667*440334654777631 is a solution to the congruence.

%t fQ[p_] := Block[{fi = First@# & /@ FactorInteger[ MultiplicativeOrder[ 10, p]]}, Union[ MemberQ[ lst, #] & /@ fi] == {True}]; k = 4; lst = {3}; While[k < 180000, If[ p = Prime@ k; fQ@ p, AppendTo[ lst, p]; Print@ p]; k++]; lst (* _Robert G. Wilson v_, Nov 30 2013 *)

%o (PARI) S=Set([3]); forprime(p=7,10^6, v=factorint(znorder(Mod(10,p)))[,1]; if(length(setintersect(S,Set(v)))==length(v), S=setunion(S,[p])) ); print(vecsort(eval(S))) \\ _Max Alekseyev_, Nov 16 2005

%Y Cf. A014950, A001270, A027889, A007138, A114207.

%K nonn

%O 1,1

%A _Vladeta Jovovic_, Dec 21 2001

%E Edited by _Max Alekseyev_, Nov 16 2005

%E Edited by _Hans Havermann_, Jul 11 2014