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

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)).

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)).

A372228 a(n) is the largest prime factor of n^n + n.

Original entry on oeis.org

2, 3, 5, 13, 313, 101, 181, 5419, 21523361, 52579, 212601841, 57154490053, 815702161, 100621, 4454215139669, 4562284561, 52548582913, 1895634885375961, 211573, 2272727294381, 415710882920521, 9299179, 1853387306082786629, 22496867303759173834520497

Offset: 1

Tyler Busby, Apr 23 2024

nonn

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

Cf. A066068, A006530, A006486, A007571, A372229.

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

from sympy import primefactors
def A372228(n): return max(max(primefactors(n),default=1),max(primefactors(n**(n-1)+1))) # Chai Wah Wu, Apr 27 2024

a(n) = A006530(A066068(n)).

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

Original entry on oeis.org

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)).

A366899 Number of prime factors of n*2^n - 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 2, 2, 2, 2, 3, 2, 4, 5, 4, 6, 3, 2, 3, 2, 4, 5, 3, 3, 2, 3, 3, 4, 5, 1, 3, 2, 3, 5, 3, 5, 2, 3, 2, 5, 4, 3, 5, 3, 4, 5, 7, 4, 4, 3, 3, 4, 5, 3, 4, 3, 4, 3, 5, 3, 3, 4, 3, 9, 6, 4, 4, 6, 4, 3, 3, 2, 5, 4, 1, 9, 3, 4, 5, 2, 1, 4, 5, 6, 2, 3, 4

Offset: 1

Tyler Busby, Oct 26 2023

nonn

The numbers n*2^n-1 are called Woodall (or Riesel) numbers.

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A001222, A003261, A085723, A366898 (divisors), A367006 (without multiplicity).

Table[PrimeOmega[n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 09 2023 *)

a(n) = bigomega(n*2^n - 1); \\ Michel Marcus, Dec 09 2023

a(n) = bigomega(n*2^n - 1) = A001222(A003261(n)).

A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

Original entry on oeis.org

1, 2, 2, 6, 4, 2, 4, 4, 4, 4, 6, 4, 12, 10, 8, 48, 8, 4, 8, 4, 16, 16, 8, 8, 4, 8, 8, 16, 24, 2, 8, 4, 8, 32, 8, 32, 4, 8, 4, 24, 16, 8, 32, 8, 16, 24, 40, 16, 16, 8, 8, 16, 24, 8, 16, 8, 16, 6, 32, 8, 8, 16, 8, 512, 48, 16, 12, 48, 16, 8, 8, 4, 24, 16, 2, 256

Offset: 1

Tyler Busby, Oct 26 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A000005, A003261, A002234, A242273, A063515, A056821, A367006, A366899.

Table[DivisorSigma[0, n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 11 2023 *)

a(n) = numdiv(n*2^n - 1); \\ Amiram Eldar, Dec 11 2023

a(n) = sigma0(n*2^n - 1) = A000005(A003261(n)).

A364818 Number of distinct prime divisors of A000129(n) (Pell numbers).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 2, 5, 2, 6, 3, 4, 2, 7, 3, 4, 4, 6, 1, 7, 3, 5, 3, 5, 3, 9, 3, 4, 4, 9, 1, 7, 2, 8, 6, 5, 2, 10, 3, 6, 5, 7, 1, 8, 5, 8, 5, 3, 1, 13, 3, 6, 6, 8, 6, 8, 2, 9, 4, 8, 3, 13, 2, 7, 8, 9, 5, 10, 4, 12, 7, 5, 2, 14, 7

Offset: 1

Tyler Busby, Oct 21 2023

nonn

			a(8)=3 because Pell(8)=408 has prime factors {2, 2, 2, 3, 17}.

Amiram Eldar, Table of n, a(n) for n = 1..630

Cf. A000129, A001221, A272040, A363829, A363831, A363833.

PrimeNu[LinearRecurrence[{2, 1}, {1, 2}, 85]] (* Amiram Eldar, Oct 21 2023 *)

a(n) = omega(Pell(n)) = A001221(A000129(n)).

A363829 Sum of the divisors of A000129(n) (Pell numbers).

Original entry on oeis.org

1, 3, 6, 28, 30, 144, 183, 1080, 1188, 3780, 5742, 52416, 33462, 131760, 251100, 1290096, 1145124, 5702400, 6804204, 42336000, 50176404, 146352096, 226041700, 2333111040, 1357893000, 4818528000, 9395060400, 47385112320, 44560482150, 251337038400, 264178169640

Offset: 1

Tyler Busby, Oct 19 2023

nonn

			a(9)=1188 because Pell(9)=985 has divisors {1, 5, 197, 985}.

Amiram Eldar, Table of n, a(n) for n = 1..630 (calculated using Jon E. Schoenfield's a-file at A000129)

Cf. A000129, A000203, A272040, A363831, A363833, A364818, A364820.

DivisorSigma[1, LinearRecurrence[{2, 1}, {1, 2}, 32]] (* Amiram Eldar, Oct 19 2023 *)

a(n) = sigma(Pell(n)) = A000203(A000129(n)).

A363831 Number of divisors of A000129(n) (Pell numbers).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 3, 16, 4, 8, 2, 72, 2, 12, 12, 40, 4, 32, 4, 96, 12, 16, 4, 384, 8, 16, 16, 144, 2, 288, 8, 96, 8, 32, 12, 1536, 8, 16, 16, 1024, 2, 288, 4, 384, 96, 32, 4, 3840, 12, 64, 32, 192, 2, 256, 32, 768, 32, 8, 2, 41472, 8, 64, 96, 896, 64, 256, 4

Offset: 1

Tyler Busby, Oct 19 2023

nonn

			a(9)=4 because Pell(9)=985 has divisors {1, 5, 197, 985}.

Amiram Eldar, Table of n, a(n) for n = 1..630 (calculated using Jon E. Schoenfield's a-file at A000129)

Cf. A000005, A000129, A272040, A363829.

DivisorSigma[0,LinearRecurrence[{2,1},{1,2},67]] (* Stefano Spezia, Oct 19 2023 *)

a(n) = sigma0(Pell(n)) = A000005(A000129(n)).

A363833 Number of prime factors of A000129(n) (Pell numbers) (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 2, 5, 2, 3, 1, 7, 1, 4, 4, 7, 2, 5, 2, 7, 4, 4, 2, 10, 3, 4, 4, 8, 1, 9, 3, 9, 3, 5, 4, 12, 3, 4, 4, 11, 1, 9, 2, 9, 7, 5, 2, 14, 4, 6, 5, 8, 1, 8, 5, 11, 5, 3, 1, 17, 3, 6, 7, 13, 6, 8, 2, 10, 4, 9, 3, 17, 2, 7, 10, 10, 6, 10, 4, 15, 7, 5, 2

Offset: 1

Tyler Busby, Oct 19 2023

nonn

			a(8)=5 because Pell(8)=408 has prime factors {2, 2, 2, 3, 17}.

Amiram Eldar, Table of n, a(n) for n = 1..630 (calculated using Jon E. Schoenfield's a-file at A000129)

Cf. A000129, A001222, A272040, A363829, A363831, A364818.

PrimeOmega[LinearRecurrence[{2,1},{1,2},83]] (* Stefano Spezia, Oct 19 2023 *)

a(n) = bigomega(Pell(n)) = A001222(A000129(n)).

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User: Tyler Busby

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

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

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A372228 a(n) is the largest prime factor of n^n + n.

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

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A366899 Number of prime factors of n*2^n - 1, counted with multiplicity.

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A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

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Mathematica

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A364818 Number of distinct prime divisors of A000129(n) (Pell numbers).

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A363829 Sum of the divisors of A000129(n) (Pell numbers).

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