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A102457 Least k >= 2 with n^(kn) == n (mod kn), also n^(kn-1) == 1 (mod k).

%I A102457 #20 Nov 14 2018 08:11:55
%S A102457 80519,2,3,2,5,2,7,2,3,2,11,2,13,2,3,2,17,2,19,2,3,2,23,2,5,2,3,2,29,
%T A102457 2,31,2,3,2,5,2,37,2,3,2,41,2,43,2,3,2,47,2,7,2,3,2,53,2,5,2,3,2,59,2,
%U A102457 61,2,3,2,5,2,67,2,3,2,71,2,73,2,3,2,7,2,79,2,3,2,83,2,5,2,3,2,89,2,7,2,3
%N A102457 Least k >= 2 with n^(kn) == n (mod kn), also n^(kn-1) == 1 (mod k).
%C A102457 Motivated by even base-2 pseudoprime 161038, I inquired into base-n pseudoprimes kn that are multiples of n, i.e., n^(kn) == n (mod kn). This is equivalent to n^(kn-1) == 1 (mod k) [_W. Edwin Clark_] and is satisfied by any k dividing n-1 [Michael Reid]. For n >= 3, this guarantees the existence of a(n) with 2 <= a(n) = k <= lpf(n-1) (lpf = least prime factor). For most n, a(n) = lpf(n-1), exceptional n and a(n) are noted in A102458 and A102459.
%H A102457 Antti Karttunen, <a href="/A102457/b102457.txt">Table of n, a(n) for n = 2..12620</a>
%H A102457 Antti Karttunen, <a href="/A102457/a102457.txt">Data supplement: n, a(n) computed for n =  2..100000</a>
%t A102457 Array[Block[{k = 2}, While[PowerMod[#, k # - 1, k] != 1, k++]; k] &, 93, 2] (* _Michael De Vlieger_, Nov 13 2018 *)
%o A102457 (PARI) A102457(n) = { for(k=2, oo, if(1==(Mod(n, k)^((k*n)-1)), return(k)); ); } \\ _Antti Karttunen_, Nov 10 2018
%Y A102457 Cf. A102458, A102459.
%Y A102457 Cf. A092067. - _R. J. Mathar_, Aug 30 2008
%K A102457 nonn
%O A102457 2,1
%A A102457 _David W. Wilson_, Jan 09 2005