A085034 -id:A085034 - cp's OEIS frontend

A057952 Number of prime factors of 9^n - 1 (counted with multiplicity).

3, 5, 5, 7, 6, 8, 5, 10, 8, 10, 7, 11, 5, 9, 11, 12, 8, 12, 7, 13, 11, 11, 6, 17, 10, 9, 13, 13, 9, 17, 8, 14, 12, 12, 11, 16, 8, 11, 15, 18, 8, 18, 6, 16, 19, 10, 10, 21, 12, 18, 15, 13, 8, 18, 15, 19, 15, 13, 7, 24, 7, 13, 19, 16, 12, 18, 8, 17, 15, 20, 9, 24, 9, 13, 22, 17, 13, 22

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690 (first 330 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A046051 (b=2), A057958 (b=3), A057957 (b=4), A057956 (b=5), A057955 (b=6), A057954 (b=7), A057953 (b=8), this sequence (b=9), A057951 (b=10), A366682 (b=11), A366708 (b=12).

Cf. A001222, A002591, A024101, A074477, A085034, A133801, A274909, A366660, A366661, A366662.

PrimeOmega[Table[9^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

Mobius transform of A085034. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024101(n)) = A057958(2*n). - Amiram Eldar, Feb 02 2020
a(n) = A057941(n) + A057958(n). - Max Alekseyev, Jan 07 2024

A085035 Number of prime factors of cyclotomic(n,10), which is A019328(n), the value of the n-th cyclotomic polynomial evaluated at x=10.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057951, number of prime factors of 10^n-1.

See references at A085021.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of Phi_n(10)

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), A085034 (x=9), this sequence (x=10).

Cf. A001222, A003060, A005422, A007138, A019328, A057951, A070528, A059892.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 10]]][[2]], {n, 1, 100}]

a(n) = A001222(A019328(n)). - Ray Chandler, May 10 2017

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.

The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008

			a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 500 term from T. D. Noe)
Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
H. Mishima, Factorization of Cyclotomic Numbers
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Eric Weisstein's World of Mathematics, Möbius Transform

omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.

Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]

a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015

A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 1, 3, 2, 1, 2, 2, 4, 1, 3, 3, 2, 2, 3, 1, 4, 3, 5, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 4, 3, 3, 2, 3, 4, 3, 2, 3, 2, 4, 2, 2, 1, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057958, number of prime factors of 3^n-1.

See references at A085021

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

omega(Phi(n,x)): A085021 (x=2), this sequence (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019321, A057958, A059885, A143663, A218356, A074477.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 3]]][[2]], {n, 1, 100}]

A085029 Number of prime factors of cyclotomic(n,4), which is A019322(n), the value of the n-th cyclotomic polynomial evaluated at x=4.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057957, number of prime factors of 4^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..1128

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), this sequence (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019322, A057957.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 4]]][[2]], {n, 1, 100}]

A085031 Number of prime factors of cyclotomic(n,6), which is A019324(n), the value of the n-th cyclotomic polynomial evaluated at x=6.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 1, 2, 2, 1, 1, 4, 1, 3, 3, 2, 2, 1, 1, 2, 3, 2, 2, 3, 3, 5, 2, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 3, 1, 3, 3, 3, 2, 2, 4, 3, 3, 3, 4, 3, 1, 4, 3, 4, 3, 2, 2, 2, 5, 1, 3, 4, 3, 3, 2, 2, 4, 3, 3, 2, 3, 7, 2, 3, 1, 4, 2, 3, 1, 2

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057955, number of prime factors of 6^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..430

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), this sequence (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019324, A057955, A059888, A274907.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 6]]][[2]], {n, 1, 100}]

A085032 Number of prime factors of cyclotomic(n,7), which is A019325(n), the value of the n-th cyclotomic polynomial evaluated at x=7.

Original entry on oeis.org

2, 3, 2, 3, 1, 1, 2, 2, 3, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 3, 2, 2, 5, 1, 3, 1, 3, 3, 3, 1, 2, 1, 3, 2, 3, 2, 3, 2, 2, 3, 2, 1, 2, 1, 4, 1, 4, 2, 3, 1, 1, 4, 4, 1, 4, 2, 4, 3, 1, 4, 5, 2, 4, 3, 3, 4, 2, 3, 5, 2, 2, 1, 3, 3, 2, 3, 5, 4, 7, 1

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057954, number of prime factors of 7^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..388

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), this sequence (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019325, A057954, A059889, A074249.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 7]]][[2]], {n, 1, 100}]

A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.

Original entry on oeis.org

2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 1, 4, 2, 4, 1, 3, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 2, 2, 4, 2, 2, 3, 2, 1, 1, 1, 2, 3, 3, 3, 4, 5, 1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 2, 5, 4, 4, 2, 2, 3, 3, 3, 5, 2, 3, 2, 3, 2, 2, 4, 3, 2, 3, 4, 3, 1, 6, 1, 2, 1, 4, 3, 4, 2, 3, 3, 4, 3

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057956, number of prime factors of 5^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..502

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), this sequence (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019323, A057956, A059887, A074479.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 5]]][[2]], {n, 1, 100}]

A085033 Number of prime factors of cyclotomic(n,8), which is A019326(n), the value of the n-th cyclotomic polynomial evaluated at x=8.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057953, number of prime factors of 8^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..500

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), this sequence (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019326, A057953, A059890, A274908.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 8]]][[2]], {n, 1, 100}]

cp's OEIS Frontend

A057952 Number of prime factors of 9^n - 1 (counted with multiplicity).

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A085035 Number of prime factors of cyclotomic(n,10), which is A019328(n), the value of the n-th cyclotomic polynomial evaluated at x=10.

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A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

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A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3.

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A085029 Number of prime factors of cyclotomic(n,4), which is A019322(n), the value of the n-th cyclotomic polynomial evaluated at x=4.

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A085031 Number of prime factors of cyclotomic(n,6), which is A019324(n), the value of the n-th cyclotomic polynomial evaluated at x=6.

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A085032 Number of prime factors of cyclotomic(n,7), which is A019325(n), the value of the n-th cyclotomic polynomial evaluated at x=7.

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A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.

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