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%I #40 May 10 2020 13:47:09
%S 3,5,6,9,10,12,13,15,16,20,21,24,30,35,39,40,45,60,63,65,80,91,105,
%T 117,120,195,240,273,315,455,585,819,1365,4095
%N 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.
%C 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.
%C The set of all numbers of this sequence is probably also finite and complete (all terms are on the list).
%C 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]
%C 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
%C No further terms below 10^5. - _M. F. Hasler_, Apr 17 2020
%t 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 *)
%o (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))<o,); n=factor(o+1)[-1..-1,1][1]; Mod(k,n)^o==1 && factor(k^o-1,n+1)[-1..-1,1][1]==n}, [1..4444]) \\ _M. F. Hasler_, Apr 17 2020
%Y Cf. A001122, A002322, A002326, A003524, A006530, A111076.
%K nonn,more
%O 1,1
%A _Thomas Ordowski_, Apr 15 2020
%E More terms from _Amiram Eldar_, Apr 15 2020