A085035 -id:A085035 - cp's OEIS frontend

A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

2, 3, 4, 4, 4, 7, 4, 6, 6, 6, 4, 9, 5, 6, 8, 8, 4, 11, 3, 9, 9, 9, 3, 12, 7, 8, 9, 10, 7, 15, 5, 13, 8, 8, 9, 14, 5, 5, 8, 13, 6, 17, 6, 13, 12, 8, 4, 15, 6, 12, 10, 11, 6, 16, 10, 14, 8, 10, 4, 22, 9, 7, 16, 17, 9, 17, 5, 12, 8, 14, 4, 20, 5, 9, 14, 8, 10, 18

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of 11...11 (Repunit).
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): this sequence (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A002283, A046053, A085035, A003020, A046107, A005422, A061075, A102380, A095370, A147556, A081317, A081318, A102347, A112505.

PrimeOmega[10^Range[70]-1] (* Jayanta Basu, May 29 2013 *)

a(n)=bigomega(10^n-1) \\ Charles R Greathouse IV, Sep 14 2015

Mobius transform of A085035 - T. D. Noe, Jun 19 2003
a(n) = Omega(10^n -1) = Omega(R_n) + 2 = A046053(n) + 2 {where Omega(n) = A001222(n) and R_n = (10^n - 1)/9 = A002275(n)}. - Lekraj Beedassy, Jun 09 2006
a(n) = A001222(A002283(n)). - Ray Chandler, Apr 22 2017

Erroneous b-file replaced by Ray Chandler, Apr 26 2017

A138940 Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

2, 4, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, 4274, 4354, 5877, 6022

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Unique period primes (A040017) are often of the form Phi(k,10) or Phi(k,-10).

Terms of this sequence which are the square of a prime, a(n)=p^2, are such that A252491(p) is prime. Apart from a(2)=2^2, there is no such term up to 26570. - M. F. Hasler, Jan 09 2015

Ray Chandler, Table of n, a(n) for n = 1..102 (first 50 terms from Robert Price, terms 92-93 from Serge Batalov, others from Kamada link)
Chris Caldwell, Unique Primes.
Makoto Kamada, Factorizations of Phi_n(10) (including prime members up to 200000).
Index entries for cyclotomic polynomials, values at X

Cf. A019328, A040017, A085035, A252491.

Cf. Subsequence of A007498, contains A004023.

Select[Range[1000], PrimeQ[Cyclotomic[#, 10]] &] (* T. D. Noe, Mar 03 2012 *)

for( i=1,999, isprime( polcyclo(i,10)) && print1( i","))

a(28)-a(43) from Robert Price, Mar 03 2012
a(44)-a(50) from Robert Price, Apr 14 2012
a(51)-a(91) from Ray Chandler, Maksym Voznyy et al. (cf. Phi_n(10) link), ca. 2009
a(92)-a(93) from Serge Batalov, Mar 28 2015

A070528 Number of divisors of 10^n-1 (999...999 with n digits).

Original entry on oeis.org

3, 6, 8, 12, 12, 64, 12, 48, 20, 48, 12, 256, 24, 48, 128, 192, 12, 640, 6, 384, 256, 288, 6, 2048, 96, 192, 96, 768, 96, 16384, 24, 6144, 128, 192, 384, 5120, 24, 24, 128, 6144, 48, 49152, 48, 4608, 1280, 192, 12, 16384, 48, 3072, 512, 1536, 48, 12288, 768

Offset: 1

Henry Bottomley, May 02 2002

			a(7)=12 since the divisors of 9999999 are 1, 3, 9, 239, 717, 2151, 4649, 13947, 41841, 1111111, 3333333, 9999999.

Table of n, a(n) for n = 1..352

Cf. A004023 (indices of 6's).
Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.
Cf. A046801, A005422, A102347, A046053, A057951, A007138, A003060, A059892, A085035.

DivisorSigma[0,#]&/@(10^Range[60]-1) (* Harvey P. Dale, Jan 14 2011 *)
Table[DivisorSigma[0, 10^n - 1], {n, 60}] (* T. D. Noe, Aug 18 2011 *)

a(n) = numdiv(10^n - 1); \\ Michel Marcus, Sep 08 2015

a(n) = A000005(A002283(n)).
a(n) = Sum_{d|n} A059892(d).
a(n) = A070529(n)*(A007949(n)+3)/(A007949(n)+1).

Terms to a(280) in b-file from Hans Havermann, Aug 19 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, May 04 2022

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

A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

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A138940 Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.

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A070528 Number of divisors of 10^n-1 (999...999 with n digits).

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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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