A344870 -id:A344870 - cp's OEIS frontend

A309941 Number of prime factors of n^n - 1, counted with multiplicity.

1, 2, 3, 4, 4, 4, 7, 8, 6, 4, 8, 6, 5, 7, 7, 7, 10, 4, 11, 10, 8, 6, 13, 13, 11, 9, 13, 10, 15, 4, 13, 12, 13, 10, 18, 11, 8, 10, 16, 9, 16, 6, 15, 18, 9, 5, 19, 20, 14, 15, 17, 8, 16, 12, 18, 10, 10, 5, 26, 8, 10, 14, 20, 19, 17, 9, 17, 12, 19, 7, 29, 15, 8, 11, 20, 13, 21, 8

Offset: 2

Hugo Pfoertner, Aug 24 2019

			a(3) = 2: 3^3 - 1 = 26 = 2 * 13.
a(5) = 4: 5^5 - 1 = 3124 = 2^2 * 11 * 71.

Amiram Eldar, Table of n, a(n) for n = 2..138 (terms 2..126 from Chai Wah Wu)
factordb, Factors of n^n-1.

Cf. A006486, A048861, A085723, A116895, A125135, A334167, A344870.

a[n_] := PrimeOmega[n^n - 1]; Array[a, 45, 2] (* Amiram Eldar, Jul 04 2024 *)

for(k=2, 50, print1(bigomega(k^k-1),", "))

A372546 Number of distinct prime factors of n^n+n.

Original entry on oeis.org

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

Offset: 1

Tyler Busby, May 06 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A001221, A066068, A372599, A372228, A344870, A344869, A377671.

a[n_] := PrimeNu[n^n + n]; Array[a, 40] (* Amiram Eldar, Oct 29 2024 *)

PARI
```
a(n) = omega(n^n+n);
```

from sympy.ntheory.factor_ import primenu
def A372546(n): return primenu(n*(n**(n-1)+1)) # Chai Wah Wu, May 07 2024

a(n) = A001221(A066068(n)).

A344869 Number of distinct prime factors of n^n+1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 3, 3, 4, 3, 6, 4, 5, 5, 6, 2, 3, 7, 5, 3, 6, 4, 8, 6, 7, 5, 11, 6, 7, 10, 7, 4, 11, 6, 13, 5, 7, 7, 8, 9, 6, 10, 8, 8, 14, 8, 10, 6, 7, 10, 11, 5, 8, 14, 11, 7, 13, 13, 9, 12, 8, 7, 18, 4, 12, 8, 7, 7, 16, 9, 8, 12, 4, 8, 24, 7, 9, 14, 7, 5, 12, 6, 12, 8, 13, 10, 12, 10, 6, 23, 15, 6, 9, 11, 16, 3, 8, 17, 23, 7

Offset: 0

Seiichi Manyama, May 31 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 0..148
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind.

Cf. A001221, A014566, A085723, A128428, A344859, A344870.

Magma
```
[#PrimeDivisors(n^n+1): n in [0..100]];
```

a[0] = 1; a[n_] := PrimeNu[n^n + 1]; Array[a, 45, 0] (* Amiram Eldar, May 31 2021 *)

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

a(n) = A001221(A014566(n)).

a(67)-a(79) from Jon E. Schoenfield, May 31 2021

a(80)-a(100) from Seiichi Manyama, May 31 2021

A372599 Number of distinct prime factors of n^n-n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 4, 6, 5, 6, 5, 9, 5, 6, 12, 8, 4, 10, 5, 11, 7, 6, 7, 12, 8, 13, 8, 10, 6, 14, 8, 12, 9, 10, 18, 18, 6, 11, 11, 19, 8, 16, 5, 12, 13, 7, 7, 20, 5, 18, 12, 14, 7, 21, 12, 19, 10, 10, 7, 24, 7, 10, 20, 15, 13, 19, 6, 19, 11, 19, 9, 25, 6, 13

Offset: 2

Tyler Busby, May 06 2024

nonn

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

Cf. A001221, A061190, A372546, A372229, A344870, A344869.

a[n_] := a[n] = PrimeNu[n^n - n];
Table[a[n], {n, 2, 75}] (* Robert P. P. McKone, May 07 2024 *)

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

from sympy.ntheory.factor_ import primenu
def A372599(n): return primenu(n*(n**(n-1)-1)) # Chai Wah Wu, May 07 2024

a(n) = A001221(A061190(n)).

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)).

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)).

A088807 Number of distinct prime factors of p^p - 1 where p = prime(n).

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 4, 3, 6, 9, 4, 9, 7, 6, 5, 7, 5, 7, 9, 7, 12, 8, 6, 8, 8, 11, 8, 6, 7, 10, 9, 7, 13

Offset: 1

Cino Hilliard, Nov 23 2003

Cf. A000040, A088730, A344870.

PrimeNu/@Table[p^p-1,{p,Prime[Range[30]]}] (* The program takes a long time to run. *) (* Harvey P. Dale, Sep 05 2020 *)

omegaptop(n,m) = { sr=0; forprime(x=2,n, y=omega(x^x-m); print1(y","); sr += 1.0/y; ); print(); }

from sympy import factorint, prime
def a(n): p = prime(n); return len(factorint(p**p-1).values())
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, May 27 2022

a(n) = A344870(A000040(n)). - Amiram Eldar, Jul 04 2024

More terms from Ray Chandler, Feb 21 2004

Name clarified, offset and data corrected and a(27)-a(33) added by Amiram Eldar, Jul 04 2024

cp's OEIS Frontend

A309941 Number of prime factors of n^n - 1, counted with multiplicity.

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A372546 Number of distinct prime factors of n^n+n.

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A344869 Number of distinct prime factors of n^n+1.

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Magma

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A372599 Number of distinct prime factors of n^n-n.

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

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

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A088807 Number of distinct prime factors of p^p - 1 where p = prime(n).

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