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A294179 a(n) is the smallest k with n prime factors such that p^k == p (mod k) for every prime p dividing k.

%I A294179 #25 Aug 03 2023 10:45:51
%S A294179 2,65,561,41041,825265,321197185,5394826801,232250619601,9746347772161
%N A294179 a(n) is the smallest k with n prime factors such that p^k == p (mod k) for every prime p dividing k.
%C A294179 All the terms are squarefree. Are all composite terms odd?
%C A294179 Conjecture: the sequence contains only finitely many Carmichael numbers, A006931. What is the smallest n >= 3 for which a(n) is not a Carmichael number? For n >= 3, a(n) <= A006931(n).
%p A294179 for k from 2 to 10^6 do
%p A294179   if numtheory:-issqrfree(k) then
%p A294179     ps := numtheory:-factorset(k);
%p A294179     n := nops(ps);
%p A294179     if not assigned(A[n]) and andmap(p -> p &^ k -p mod k = 0, ps) then
%p A294179       A[n] := k;
%p A294179     end if
%p A294179   end if;
%p A294179 end do:
%p A294179 seq(A[i],i=1..max(map(op, [indices(A)]))); # _Robert Israel_, Feb 11 2018
%t A294179 With[{s = Select[Range[10^6], Function[k, AllTrue[FactorInteger[k][[All, 1]], PowerMod[#, k, k] == Mod[#, k] &]]]}, Select[Table[SelectFirst[s, PrimeOmega@ # == n &], {n, 5}], IntegerQ]] (* _Michael De Vlieger_, Feb 20 2018 *)
%Y A294179 Cf. A006931, A285549, A294169.
%K A294179 nonn,more,hard
%O A294179 1,1
%A A294179 _Thomas Ordowski_, Feb 11 2018
%E A294179 a(7)-a(8) from _Daniel Suteu_, Feb 06 2023
%E A294179 a(9) from _Michael S. Branicky_, Aug 03 2023