A290532 Irregular triangle read by rows in which row n lists the number of divisors of each divisor of n.

1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 2, 4, 1, 2, 1, 2, 2, 3, 4, 6, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 2, 4, 3, 6, 1, 2, 1, 2, 3, 2, 4, 6, 1, 2, 2, 4, 1, 2, 2, 4, 1, 2, 1, 2, 2, 3, 4, 4, 6, 8, 1, 2, 3

Offset: 1

Author

Michel Lagneau, Aug 05 2017

Keywords

• nonn
• tabf

Comments

Or, in the triangle A027750, replace each element with the number of its divisors.

The row of index n = p^m (p prime and m >= 0) is equal to (1, 2, ..., m + 1);

We observe an interesting property when the index n of the row n is the product of k distinct primes, k = 1,2,... For example:

The index n is prime => row n = (1, 2);

The index n equals the product of two distinct primes => row n = (1, 2, 2, 4);

The index n equals the product of three distinct primes => row n = (1, 2, 2, 2, 4, 4, 4, 8) or a permutation of the elements;

...

Let us now consider Pascal's triangle (A007318(n) for n > 0):

1, 1;

1, 2, 1;

1, 3, 3, 1;

1, 4, 6, 4, 1;

...

Row 1 of Pascal's triangle gives the number of "1" and the number of "2" respectively belonging to the row of index n = prime(m) of the sequence;

Row 2 of Pascal's triangle gives the number of "1", the number of "2" and the number of "4" respectively belonging to the row of index n = p*q of the sequence, where p and q are distinct primes;

Row 3 of Pascal's triangle gives the number of "1", the number of "2", the number of "4" and the number of "8" respectively belonging to the row of index n = p*q*r of the sequence, where p, q and r are distinct primes;

...

It is now easy to generalize this process by the following proposition.

Proposition: binomial(m,k) is the number of terms of the form 2^k belonging to the row of index n in the sequence when n is the product of m distinct primes.

Examples

			Row 6 is (1, 2, 2, 4) because the 6th row of A027750 is [1, 2, 3, 6] and tau(1) = 1, tau(2) = 2, tau(3) = 2 and tau(6) = 4.
Triangle begins:
  1;
  1, 2;
  1, 2;
  1, 2, 3;
  1, 2;
  1, 2, 2, 4;
  1, 2;
  1, 2, 3, 4;
  1, 2, 3;
  1, 2, 2, 4;
  ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened)

Crossrefs

Cf. A000005, A007318, A027750, A084997, A290478.

Programs

Maple

with(numtheory):nn:=100:
for n from 1 to nn do:
  d1:=divisors(n):n1:=nops(d1):
   for i from 1 to n1 do:
     n2:=tau(d1[i]):
     printf(`%d, `,n2):
   od:
od:

Mathematica

Table[DivisorSigma[0, Divisors@ n], {n, 25}] // Flatten (* Michael De Vlieger, Aug 07 2017 *)

PARI

row(n) = apply(numdiv, divisors(n)); \\ Michel Marcus, Dec 27 2021

Formula

T(n, k) = tau(A027750(n, k)).