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A062319
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Number of divisors of n^n, or of A000312(n).
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28
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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
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OFFSET
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0,3
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COMMENTS
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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)
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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Join[{1}, DivisorSigma[0, #^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)
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PROG
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(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) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021
(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
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CROSSREFS
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Taking Omega instead of sigma gives A066959.
Diagonal n = k of the array A343656.
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.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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