A102347 -id:A102347 - cp's OEIS frontend

A366651 Number of distinct prime divisors of 8^n - 1.

1, 2, 2, 4, 3, 4, 3, 6, 3, 6, 4, 8, 4, 6, 6, 9, 5, 6, 4, 11, 6, 8, 4, 12, 7, 7, 6, 12, 6, 11, 3, 12, 8, 10, 10, 12, 6, 8, 9, 15, 5, 11, 5, 14, 10, 8, 6, 17, 5, 13, 8, 16, 8, 12, 10, 17, 7, 10, 6, 21, 5, 7, 9, 15, 8, 15, 6, 19, 9, 20, 7, 18, 7, 12, 14, 16, 9

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

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

Cf. A024088, A001221, A057953, A366652, A366653, A366654.

Cf. A046800, A133801, A366604, A366611, A366620, A366632, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(8^n - 1), ", "))

a(n) = omega(8^n-1) = A001221(A024088(n)).

a(n) = A046800(3*n). - Max Alekseyev, Jan 09 2024

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

13, 156, 1520, 15912, 148512, 2042880, 14508000, 162493344, 1534205464, 16203253248, 144451398000, 2063316971520, 14903272088640, 158269280832000, 1614847741624320, 17205180696931968, 144444514193267496

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (terms 1..322 from Ray Chandler)
C. Caldwell, Sigma function.

Cf. A000203, A001270, A002283, A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102347, A102380, A112505, A147556, A295503, A366669.

DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)

a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A000203(A002283(n)). - Ray Chandler, Apr 22 2017

A112505 Number of primitive prime factors of 10^n-1.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Sep 08 2005

Also the number of primes whose reciprocal is a repeating decimal of length n. The number of numbers in each row of table A046107.

By Zsigmondy's theorem, a(n) >= 1. When a(n)=1, the corresponding prime is called a unique prime (see A007498, A040017 and A051627).

Table of n, a(n) for n = 1..352
Eric Weisstein's World of Mathematics, Primitive Prime Factor
Eric Weisstein's World of Mathematics, Zsigmondy Theorem
Eric Weisstein's World of Mathematics, Unique Prime

Cf. A007138 (smallest primitive prime factor of 10^n-1), A102347 (number of distinct prime factors of 10^n-1), A046107.

pp={}; Table[f=Transpose[FactorInteger[10^n-1]][[1]]; p=Complement[f, pp]; pp=Union[pp, p]; Length[p], {n, 66}]

Terms to a(276) in b-file from T. D. Noe, Jun 01 2010
a(277)-a(322) in b-file from Ray Chandler, May 01 2017
a(323)-a(352) in b-file from Max Alekseyev, Apr 28 2022

A366611 Number of distinct prime divisors of 5^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 5, 2, 6, 2, 5, 6, 6, 3, 7, 4, 8, 5, 6, 3, 8, 7, 5, 8, 7, 4, 11, 3, 8, 5, 6, 8, 11, 4, 8, 5, 11, 3, 12, 3, 9, 11, 6, 2, 11, 3, 11, 7, 8, 4, 14, 8, 9, 6, 7, 3, 17, 4, 7, 10, 11, 7, 12, 6, 11, 8, 14, 3, 16, 4, 8, 15, 11, 6, 11, 4, 15

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

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

Cf. A024049, A001221, A057956, A059887, A074479, A143665, A218357, A295502, A366612, A366613.

Cf. A046800, A133801, A366604, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(5^n - 1), ", "))

a(n) = omega(5^n-1) = A001221(A024049(n)).

A087021 Number of distinct prime factors of n-th cyclic number.

Original entry on oeis.org

4, 8, 9, 8, 10, 8, 10, 21, 23, 19, 19, 15, 16, 12, 11, 33, 31, 19, 24, 22, 24, 18, 14, 33, 39, 23, 36, 13, 13, 19, 36, 32, 29, 27, 25, 11, 20, 56, 37, 46, 25, 22, 21, 16, 47, 25, 33, 22, 55, 32, 25

Offset: 1

Reinhard Zumkeller, Jul 30 2003

A004042(n) factorized with Dario Alpern's ECM.

Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

			A004042(2) = 142857 = 37*13*11*3^3, therefore a(1) =
#{3,11,13,37} = 4.

Dario A. Alpern, Factorization using the Elliptic Curve Method.
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Cyclic Number

Cf. A001913, A087021-A087026.

a(n) = A001221(A004042(n+1)).

For n>1, let p = A001913(n). If p is a base-10 Wieferich prime, then a(n) = A102347(p-1) + 2; otherwise a(n) = A102347(p-1) + 1. Also, we have A102347(p-1) = A102347((p-1)/2) + A119704((p-1)/2). - Max Alekseyev, Apr 26 2022

a(3) corrected, a(12)-a(42) added by Ray Chandler, Nov 16 2011

a(43)-a(51) from Max Alekseyev, May 13 2022

A366921 a(n) is the least prime factor > 3 of 10^n - 1.

Original entry on oeis.org

11, 37, 11, 41, 7, 239, 11, 37, 11, 21649, 7, 53, 11, 31, 11, 2071723, 7, 1111111111111111111, 11, 37, 11, 11111111111111111111111, 7, 41, 11, 37, 11, 3191, 7, 2791, 11, 37, 11, 41, 7, 2028119, 11, 37, 11, 83, 7, 173, 11, 31, 11, 35121409, 7, 239, 11, 37, 11, 107

Offset: 2

Hugo Pfoertner, Oct 28 2023

nonn

			a(2) = 11 because 99 = 3^2 * 11 = 3^A366922(2) * A003020(2).
a(9) = 37 because 10^9 - 1 = 3^4 * 37 * 333667 = 3^A366922(9) * 37 * A003020(9).

Cf. A002283, A003020, A102347, A366922.

PARI
```
a366921(n) = factor(10^n-1)[2,1]
```

A366922 a(n) is the exponent of 3 in the prime factorization of 10^n - 1.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Oct 28 2023

1

Cf. A002283, A003020, A007949, A051064, A057951, A102347, A366921.

a[n_]:=IntegerExponent[10^n-1,3]; Array[a,90] (* Stefano Spezia, Oct 28 2023 *)

PARI
```
a366922(n) = valuation(10^n-1,3)
```

def A366922(n):
    c = 0
    a, b = divmod(10**n-1, 3)
    while b == 0:
        a, b = divmod(a, 3)
        c += 1
    return c # Chai Wah Wu, Oct 29 2023

a(n) = A007949(10^n - 1).

a(n) = A007949(n) + 2 = A051064(n) + 1.

cp's OEIS Frontend

A366651 Number of distinct prime divisors of 8^n - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A112505 Number of primitive prime factors of 10^n-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A366611 Number of distinct prime divisors of 5^n - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A087021 Number of distinct prime factors of n-th cyclic number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A366921 a(n) is the least prime factor > 3 of 10^n - 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A366922 a(n) is the exponent of 3 in the prime factorization of 10^n - 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

Formula