A372228 -id:A372228 - cp's OEIS frontend

A372229 a(n) is the largest prime factor of n^n - n.

2, 3, 7, 13, 311, 43, 337, 193, 333667, 13421, 266981089, 28393, 29914249171, 10678711, 1321, 184417, 7563707819165039903, 236377, 192696104561, 920421641, 12271836836138419, 39700406579747, 58769065453824529, 152587500001, 4315817869647001, 797161

Offset: 2

Tyler Busby, Apr 23 2024

nonn

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

Cf. A061190, A006530, A006486, A007571, A372228.

pf := n -> NumberTheory:-PrimeFactors(n): a := n -> max(pf(n^n - n));
seq(a(n), n = 2..27);  # Peter Luschny, Apr 27 2024

Table[f = FactorInteger[n^n-n]; f[[Length[f]]][[1]], {n,2,25}] (* Vaclav Kotesovec, Apr 26 2024 *)

from sympy import primefactors
def A372229(n): return max(max(primefactors(n),default=1),max(primefactors(n**(n-1)-1),default=1)) # Chai Wah Wu, Apr 27 2024

a(n) = A006530(A061190(n)).

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)).

A377671 Number of prime factors of n^n+n (counted with multiplicity).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

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

Cf. A001222, A066068, A085723, A372228, A372546, A377672, A377673, A377674.

seq(numtheory:-bigomega(n^n+n),n=1..76); # Robert Israel, Nov 03 2024

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

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

a(n) = A001222(A066068(n)).

A377672 a(n) is the number of divisors of n^n + n.

Original entry on oeis.org

2, 4, 8, 12, 8, 32, 48, 48, 12, 128, 16, 24, 16, 256, 64, 80, 32, 96, 1536, 96, 64, 1024, 64, 96, 96, 512, 512, 3072, 64, 128, 2048, 384, 128, 8192, 128, 1152, 256, 128, 32, 2048, 128, 128, 6144, 288, 768, 262144, 64, 480, 1536, 1536, 2048, 3072, 1024, 1024

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

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

Cf. A000005, A066068, A344859, A372228, A372546, A377671, A377673, A377674.

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

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

a(n) = A000005(A066068(n)).

A377673 a(n) is the sum of the divisors of n^n + n.

Original entry on oeis.org

3, 12, 72, 588, 5652, 117504, 1895712, 46503600, 839411118, 25440307200, 474527311344, 22404560101168, 489294047662728, 30902868417576960, 1096805935992340800, 38000593697802058224, 1318965178069293272496, 90596485743469636057920, 3578317312662511683264000

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

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

Cf. A000203, A066068, A366820, A372228, A372546, A377671, A377672, A377674.

a[n_] := DivisorSigma[1, n^n + n]; Array[a, 20] (* Amiram Eldar, Nov 04 2024 *)

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

a(n) = A000203(A066068(n)).

A377674 a(n) = phi(n^n + n) where phi is the Euler totient function.

Original entry on oeis.org

1, 2, 8, 96, 1248, 12000, 259200, 5461344, 129140160, 2725643520, 127561104000, 2743415522496, 139778722137600, 2504616361228800, 111747349423990784, 8644660582219776000, 387774574486565683200, 12306643656809728412160, 816897235219321957908480

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

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

Cf. A000010, A066068, A366822, A372228, A372546, A377671, A377672, A377673.

a[n_] := EulerPhi[n^n + n]; Array[a, 20] (* Amiram Eldar, Nov 04 2024 *)

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

a(n) = A000010(A066068(n)).

cp's OEIS Frontend

A372229 a(n) is the largest prime factor of n^n - n.

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

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A377671 Number of prime factors of n^n+n (counted with multiplicity).

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A377672 a(n) is the number of divisors of n^n + n.

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A377673 a(n) is the sum of the divisors of n^n + n.

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Mathematica

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Formula

A377674 a(n) = phi(n^n + n) where phi is the Euler totient function.

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