A062319 Number of divisors of n^n, or of A000312(n).

1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, 14, 225, 256, 65, 18, 703, 20, 861, 484, 529, 24, 1825, 51, 729, 82, 1653, 30, 29791, 32, 161, 1156, 1225, 1296, 5329, 38, 1521, 1600, 4961, 42, 79507, 44, 4005, 4186, 2209, 48, 9457, 99, 5151, 2704, 5565, 54

Offset: 0

Author

Jason Earls, Jul 05 2001

Keywords

• easy
• nonn

Comments

From Gus Wiseman, May 02 2021: (Start)

Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. For example, the a(1) = 1 through a(5) = 6 tuples are:

(1) (1,1) (1,1,1) (1,1,1,1) (1,1,1,1,1)

(1,2) (1,1,3) (1,1,1,2) (1,1,1,1,5)

(2,1) (1,3,1) (1,1,1,4) (1,1,1,5,1)

(3,1,1) (1,1,2,1) (1,1,5,1,1)

(1,1,4,1) (1,5,1,1,1)

(1,2,1,1) (5,1,1,1,1)

(1,4,1,1)

(2,1,1,1)

(4,1,1,1)

The unordered case (pairwise coprime n-multisets of divisors of n) is counted by A343654.

(End)

Examples

			From _Gus Wiseman_, May 02 2021: (Start)
The a(1) = 1 through a(5) = 6 divisors:
  1  1  1   1    1
     2  3   2    5
     4  9   4    25
        27  8    125
            16   625
            32   3125
            64
            128
            256
(End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Crossrefs

Cf. A046798, A077592, A035116. - Enrique Pérez Herrero, Nov 09 2010

Number of divisors of A000312(n).

Taking Omega instead of sigma gives A066959.

Positions of squares are A173339.

Diagonal n = k of the array A343656.

A000005 counts divisors.

A059481 counts k-multisets of elements of {1..n}.

A334997 counts length-k strict chains of divisors of n.

A343658 counts k-multisets of divisors.

Pairwise coprimality:

- A018892 counts coprime pairs of divisors.

- A084422 counts pairwise coprime subsets of {1..n}.

- A100565 counts pairwise coprime triples of divisors.

- A225520 counts pairwise coprime sets of divisors.

- A343652 counts maximal pairwise coprime sets of divisors.

- A343653 counts pairwise coprime non-singleton sets of divisors > 1.

- A343654 counts pairwise coprime sets of divisors > 1.

Cf. A000169, A000272, A002064, A002109, A009998, A048691, A143773, A146291, A176029, A327527, A343657.

Programs

Magma

[NumberOfDivisors(n^n): n in  [0..60]]; // Vincenzo Librandi, Nov 09 2014

Mathematica

A062319[n_IntegerQ]:=DivisorSigma[0,n^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
Join[{1},DivisorSigma[0,#^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)

PARI

je=[]; for(n=0,200,je=concat(je,numdiv(n^n))); je

PARI

{ for (n=0, 1000, write("b062319.txt", n, " ", numdiv(n^n)); ) } \\ Harry J. Smith, Aug 04 2009

PARI

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,2]*n+1) \\ Franklin T. Adams-Watters, May 03 2011

PARI

a(n) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021

Python

from math import prod
from sympy import factorint
def A062319(n): return prod(n*d+1 for d in factorint(n).values()) # Chai Wah Wu, Jun 03 2021

Formula

a(n) = A000005(A000312(n)). - Enrique Pérez Herrero, Nov 09 2010

a(2^n) = A002064(n). - Gus Wiseman, May 02 2021

a(prime(n)) = prime(n) + 1. - Gus Wiseman, May 02 2021

a(n) = Product_{i=1..s} (1 + n * m_i) where (m_1,...,m_s) is the sequence of prime multiplicities (prime signature) of n. - Gus Wiseman, May 02 2021

a(n) = Sum_{d|n} n^omega(d) for n > 0. - Seiichi Manyama May 12 2021