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A334133
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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