A309941 -id:A309941 - cp's OEIS frontend

A344870 Number of distinct prime factors of n^n-1.

1, 2, 3, 3, 4, 4, 6, 6, 5, 4, 8, 5, 5, 7, 7, 4, 8, 3, 11, 9, 8, 6, 12, 11, 8, 9, 11, 9, 14, 4, 12, 8, 13, 10, 18, 9, 8, 10, 15, 7, 16, 6, 14, 17, 8, 5, 18, 17, 13, 14, 17, 7, 15, 10, 18, 8, 10, 5, 26, 7, 9, 14, 19, 14, 17, 9, 15, 11, 19, 7, 29, 12, 7, 11, 19, 12, 21, 8, 22, 25, 6, 6, 26, 16, 9, 15, 21, 8, 26, 11, 15, 13, 11, 11, 25, 8, 12, 14, 26

Offset: 2

Seiichi Manyama, May 31 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 2..138
factordb, Factors of n^n-1.

Cf. A001221, A048861, A309941, A334167, A344869.

Magma
```
[#PrimeDivisors(n^n-1): n in [2..100]];
```

a[n_] := PrimeNu[n^n - 1]; Array[a, 45, 2] (* Amiram Eldar, Jun 01 2021 *)

PARI
```
a(n) = omega(n^n-1);
```

a(n) = A001221(A048861(n)).

A085723 Number of prime divisors of n^n+1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 5, 3, 4, 3, 7, 4, 5, 6, 9, 2, 4, 9, 6, 3, 6, 4, 10, 6, 7, 5, 12, 6, 7, 10, 11, 4, 11, 6, 15, 5, 7, 8, 10, 10, 6, 10, 9, 8, 14, 8, 13, 6, 8, 10, 12, 5, 10, 14, 13, 7, 13, 13, 10, 12, 8, 7, 24, 4, 12, 8, 8, 7, 17, 10, 11, 12, 4, 8, 25, 7, 9, 14, 10, 5, 12, 7, 13, 8

Offset: 1

Jason Earls, Jul 20 2003

16^16+1 = 274177 * 67280421310721 is a semiprime. Where is the next?
a(73) >= 4. - Donovan Johnson, Sep 27 2010
According to factordb there are currently no other known candidates for semiprimes, with 781^781+1 being the largest fully factored number of this form. - Hugo Pfoertner, Aug 24 2019

			a(3) = 3: 3^3 + 1 = 28 = 2^2 * 7.
a(4) = 1: 4^4 + 1 = 257 is prime.
a(5) = 3: 5^5 + 1 = 3126 = 2 * 3 * 521.

Amiram Eldar, Table of n, a(n) for n = 1..148 (terms 1..123 from Chai Wah Wu)
factordb, Factors of n^n+1.

Cf. A001222, A007571, A014566, A055385, A121270, A309941, A344869.

for(k=1, 60, print1(bigomega(k^k+1),", ")) \\ Hugo Pfoertner, Aug 24 2019

a(n) = A001222(A014566(n)). - Amiram Eldar, Sep 27 2024

More terms from Ray G. Opao, Aug 25 2004
Corrected 8 existing terms and a(46)-a(72) from Donovan Johnson, Sep 27 2010
a(73)-a(84) added by Hugo Pfoertner, Aug 24 2019

A366819 a(n) is the sum of the divisors of n^n-1.

Original entry on oeis.org

4, 42, 432, 6048, 67584, 1704240, 38054016, 967814400, 16203253248, 513593801496, 15743437516800, 720045832568832, 19146847615988736, 835966563470742528, 31421980989189888768, 1602925310146310674200, 52064744760120508416000, 4286575920597346109768658

Offset: 2

Sean A. Irvine, Oct 24 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 2..138

Cf. A000203, A048861, A309941, A344870, A334167, A062727, A366820.

Array[DivisorSigma[1, #^# - 1] &, 18, 2] (* Michael De Vlieger, Oct 24 2023 *)

PARI
```
a(n) = sigma(n^n-1);
```

a(n) = A000203(A048861(n)).

A377675 Number of prime factors of n^n-n (counted with multiplicity).

Original entry on oeis.org

1, 4, 5, 7, 5, 9, 7, 12, 8, 9, 7, 13, 6, 11, 17, 16, 6, 17, 7, 15, 10, 10, 10, 19, 11, 18, 15, 14, 7, 22, 13, 21, 11, 14, 22, 24, 7, 15, 15, 26, 9, 20, 7, 17, 17, 12, 11, 30, 9, 24, 15, 20, 10, 29, 16, 27, 12, 13, 9, 29, 8, 18, 29, 27, 15, 24, 8, 23, 13, 25

Offset: 2

Sean A. Irvine, Nov 03 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 2..115

Cf. A001222, A061190, A309941, A372229, A372599, A377671, A377676, A377677, A377678.

a[n_] := PrimeOmega[n^n - n]; Array[a, 45, 2] (* Amiram Eldar, Nov 04 2024 *)

PARI
```
a(n) = bigomega(n^n-n);
```

a(n) = A001222(A061190(n)).

A366821 a(n) is phi(n^n-1) where phi is the Euler totient function.

Original entry on oeis.org

2, 12, 128, 1400, 30240, 264992, 6635520, 141087744, 5890320000, 114117380608, 4662793175040, 99053063903040, 5470524984113280, 167080949856000000, 9208981628670443520, 413582117375670921216, 29531731481729468006400, 659473218553437863041320

Offset: 2

Sean A. Irvine, Oct 24 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 2..138 (terms 2..102 from Robert G. Wilson v)

Cf. A000010, A048861, A064447, A309941, A334167, A344870, A366819, A366822.

a:= n-> numtheory[phi](n^n-1):
seq(a(n), n=2..20);  # Alois P. Heinz, Oct 26 2023

Array[EulerPhi[#^# - 1] &, 18, 2] (* Michael De Vlieger, Oct 24 2023 *)

PARI
```
a(n) = eulerphi(n^n-1);
```

a(n) = A000010(A048861(n)).

cp's OEIS Frontend

A344870 Number of distinct prime factors of n^n-1.

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A085723 Number of prime divisors of n^n+1 (counted with multiplicity).

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A366819 a(n) is the sum of the divisors of n^n-1.

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A377675 Number of prime factors of n^n-n (counted with multiplicity).

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A366821 a(n) is phi(n^n-1) where phi is the Euler totient function.

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