A274909 -id:A274909 - cp's OEIS frontend

A085034 Number of prime factors of cyclotomic(n,9), which is A019327(n), the value of the n-th cyclotomic polynomial evaluated at x=9.

3, 2, 2, 2, 3, 1, 2, 3, 3, 2, 4, 1, 2, 2, 3, 2, 5, 1, 4, 1, 4, 2, 3, 3, 4, 2, 5, 2, 6, 1, 5, 2, 3, 2, 3, 1, 5, 2, 8, 2, 5, 2, 3, 3, 5, 2, 7, 2, 7, 4, 5, 2, 5, 1, 5, 3, 6, 2, 4, 3, 4, 3, 5, 2, 4, 1, 5, 3, 7, 3, 6, 2, 6, 3, 7, 4, 4, 2

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057952, number of prime factors of 9^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..690

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), this sequence (x=9), A085035 (x=10).

Cf. A019327, A057952, A059891, A274909.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 9]]][[2]], {n, 1, 100}]

A366718 Largest prime factor of 12^n - 1.

Original entry on oeis.org

11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.

[Maximum(PrimeDivisors(12^n-1)): n in [1..40]];

Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024140(n)).

A002591 Largest prime factor of 3^(2n+1) - 1.

Original entry on oeis.org

2, 13, 11, 1093, 757, 3851, 797161, 4561, 34511, 363889, 368089, 1001523179, 391151, 8209, 20381027, 4404047, 2413941289, 2664097031, 17189128703, 797161, 86950696619, 380808546861411923, 927001, 96656723, 131713, 99810171997

Offset: 0

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 28.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 0..344
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Together with A274909 forms bisection of A074477.Cf. A057958, A059885, A085028, A133801, A235366.

Table[FactorInteger[3^(2n-1)-1][[-1,1]],{n,30}] (* Harvey P. Dale, Oct 19 2022 *)

a(n) = A074477(2n+1). - Max Alekseyev, May 22 2022

Corrected and extended by Jud McCranie, Jan 03 2001
Terms up to a(307) in b-file from Sean A. Irvine, Apr 20 2014
a(0) prepended and a(308)-a(344) added to b-file by Max Alekseyev, Apr 24 2019, Sep 10 2020, Aug 26 2021, May 22 2022

cp's OEIS Frontend

A085034 Number of prime factors of cyclotomic(n,9), which is A019327(n), the value of the n-th cyclotomic polynomial evaluated at x=9.

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A366718 Largest prime factor of 12^n - 1.

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A002591 Largest prime factor of 3^(2n+1) - 1.

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