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

2, 65, 561, 41041, 825265, 321197185, 5394826801, 232250619601, 9746347772161

Offset: 1

Author

Thomas Ordowski, Feb 11 2018

Keywords

• nonn
• more
• hard

Comments

All the terms are squarefree. Are all composite terms odd?

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).

Crossrefs

Cf. A006931, A285549, A294169.

Programs

Maple

for k from 2 to 10^6 do
  if numtheory:-issqrfree(k) then
    ps := numtheory:-factorset(k);
    n := nops(ps);
    if not assigned(A[n]) and andmap(p -> p &^ k -p mod k = 0, ps) then
      A[n] := k;
    end if
  end if;
end do:
seq(A[i],i=1..max(map(op, [indices(A)]))); # Robert Israel, Feb 11 2018

Mathematica

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 *)

Extensions

a(7)-a(8) from Daniel Suteu, Feb 06 2023

a(9) from Michael S. Branicky, Aug 03 2023