OFFSET
0,3
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)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)
FORMULA
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
EXAMPLE
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)
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 *)
PROG
(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
(Magma) [NumberOfDivisors(n^n): n in [0..60]]; // Vincenzo Librandi, Nov 09 2014
(Python 3.8+)
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
CROSSREFS
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.
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Jul 05 2001
STATUS
approved