A334167 -id:A334167 - 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)).

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

Original entry on oeis.org

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),", "))

A344859 a(n) is the number of divisors of n^n + 1.

Original entry on oeis.org

2, 2, 2, 6, 2, 8, 8, 16, 8, 16, 8, 96, 16, 32, 48, 160, 4, 12, 288, 48, 8, 64, 16, 512, 64, 128, 32, 3072, 64, 128, 1024, 384, 16, 2048, 64, 18432, 32, 128, 192, 512, 768, 64, 1024, 384, 256, 16384, 256, 2560, 64, 192, 1024, 3072, 32, 512, 16384, 4096, 128, 8192, 8192, 768, 4096, 256, 128, 1376256, 16

Offset: 0

Seiichi Manyama, May 31 2021

nonn

Tyler Busby, Table of n, a(n) for n = 0..148 (terms 0..123 from Max Alekseyev)
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000005, A014566, A062319, A064144, A121270, A334167.

Cf. A007571 (Lpf), A055385 (lpf).

a[0] = 2; a[n_] := DivisorSigma[0, n^n + 1]; Array[a, 45, 0] (* Amiram Eldar, May 31 2021 *)

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

a(n) = A000005(A014566(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)).

A377676 a(n) is the number of divisors of n^n - n.

Original entry on oeis.org

2, 8, 18, 40, 24, 120, 48, 336, 80, 192, 72, 1920, 48, 288, 23040, 1728, 36, 10240, 72, 7680, 432, 240, 384, 32256, 640, 49152, 2016, 3840, 96, 193536, 1152, 22528, 1152, 4608, 1327104, 1638400, 96, 7680, 9216, 4128768, 384, 294912, 72, 23040, 30720, 576

Offset: 2

Sean A. Irvine, Nov 03 2024

nonn

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

Cf. A000005, A061190, A334167, A372229, A372599, A377672, A377675, A377677, A377678.

a[n_] := DivisorSigma[0, n^n - n]; Array[a, 45, 2] (* Amiram Eldar, Nov 04 2024 *)

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

a(n) = A000005(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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A309941 Number of prime factors of n^n - 1, counted with multiplicity.

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A344859 a(n) is the number of divisors of n^n + 1.

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

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A377676 a(n) is the number of divisors of n^n - n.

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Mathematica

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

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