A334133 Numbers k > 2 such that gpf(A111076(k)^lambda(k) - 1) = gpf(lambda(k) + 1); where gpf is the greatest prime factor (A006530), and lambda = A002322 is the Carmichael function.

3, 5, 6, 9, 10, 12, 13, 15, 16, 20, 21, 24, 30, 35, 39, 40, 45, 60, 63, 65, 80, 91, 105, 117, 120, 195, 240, 273, 315, 455, 585, 819, 1365, 4095

Offset: 1

Author

Thomas Ordowski, Apr 15 2020

Keywords

• nonn
• more

Comments

Prime numbers in this sequence are 3, 5, and 13. These are primes p with primitive root 2 (A001122) such that gpf(2^(p-1)-1) = p.

The set of all numbers of this sequence is probably also finite and complete (all terms are on the list).

The odd terms of this sequence up to 4095 = 2^12-1 are exactly the divisors of this number (A003524) except 1 and 7. [Edited by M. F. Hasler, Apr 17 2020]

Conjecture: all odd terms {3, 5, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095} are odd numbers k such that gpf(2^m-1) = gpf(m+1), where m = A002326((k-1)/2) is the multiplicative order of 2 mod 2k+1. - Amiram Eldar, Apr 15 2020

No further terms below 10^5. - M. F. Hasler, Apr 17 2020

Crossrefs

Cf. A001122, A002322, A002326, A003524, A006530, A111076.

Programs

Mathematica

gpf[n_] := FactorInteger[n][[-1, 1]]; gpfQ[n_, p_] := Module[{ps = Select[Range[p], PrimeQ], np, m, k}, np = Length[ps]; m = n; If[Divisible[n, p], Do[m /= (ps[[k]]^IntegerExponent[m, ps[[k]]]), {k, 1, np}]; m == 1, False]]; f[n_] := Module[{k = 1, lam = CarmichaelLambda[n]}, While[! CoprimeQ[n, k] || MultiplicativeOrder[k, n] != lam, k++]; k]; Select[Range[2, 2^12], gpfQ[f[#]^(c = CarmichaelLambda[#]) - 1, gpf[c + 1]] &] (* Amiram Eldar, Apr 15 2020 *)

PARI

select( is_A334133(n)={n>2||return; my(o=lcm(znstar(n)[2]),k=1); while(gcd(k++,n)>1 || znorder(Mod(k,n))M. F. Hasler, Apr 17 2020

Extensions

More terms from Amiram Eldar, Apr 15 2020