A079025 Triangular array read by rows: column sums of frequency distributions associated with number of divisors of least prime signatures.

1, 1, 1, 2, 3, 2, 3, 6, 6, 3, 5, 12, 16, 12, 5, 7, 20, 32, 32, 20, 7, 11, 35, 65, 79, 65, 35, 11, 15, 54, 113, 160, 160, 113, 54, 15, 22, 86, 199, 318, 371, 318, 199, 86, 22, 30, 128, 323, 573, 756, 756, 573, 323, 128, 30, 42, 192, 523, 1013, 1485, 1683, 1485, 1013, 523, 192, 42

Offset: 0

Author

Alford Arnold, Feb 01 2003

Keywords

• easy
• nonn
• tabl

Comments

Row sums of the triangular table is sequence A074141. The left column and the main diagonal are the partition numbers A000041.

T(n,k) is the total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of k and m runs through the set of least numbers whose prime signature is a partition of n. - Alois P. Heinz, Aug 23 2019

Examples

			The seven least integers associated with prime signatures 5, 41, 32, 311, 221, 2111, 11111 (partitions of 5) are 32, 48, 72, 120, 180, 420 and 2310 (see A036035).  The corresponding numbers of divisors 6, 10, 12, 16, 18, 24 and 32 (see A074139) can be refined with the following frequency distributions D(p,s), which counts how many divisors of the entry of A036035 have a sum of prime exponents s, 0<=s<=n:
  1  1  1  1  1 1
  1  2  2  2  2 1
  1  2  3  3  2 1
  1  3  4  4  3 1
  1  3  5  5  3 1
  1  4  7  7  4 1
  1  5 10 10  5 1 , therefore the column sums are:
  7 20 32 32 20 7 , which is row 5 of the triangle.
Triangle T(n,k) begins:
    1
    1   1
    2   3    2
    3   6    6    3
    5  12   16   12    5
    7  20   32   32   20     7
   11  35   65   79   65    35    11
   15  54  113  160  160   113    54    15
   22  86  199  318  371   318   199    86    22
   30 128  323  573  756   756   573   323   128   30
   42 192  523 1013 1485  1683  1485  1013   523  192   42
   56 275  803 1683 2701  3405  3405  2701  1683  803  275   56
   77 399 1237 2776 4822  6662  7413  6662  4822 2776 1237  399  77
  101 556 1826 4366 8144 12205 14901 14901 12205 8144 4366 1826 556 101
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Columns k=0-10 give: A000041, A006128, A309691, A309693, A309919, A309920, A309921, A309922, A309923, A309924, A309925.

Row sums give A074141.

T(2n,n) gives A309915.

Cf. A036035, A074139, A079474, A087443.

Programs

Maple

A079025 := proc(n,k)
    local psig ,d,a;
    a := 0 ;
    for psig in A036035_row(n) do
        for d in numtheory[divisors](psig) do
            if numtheory[bigomega](d) = k then
                a := a+1 ;
            end if:
        end do:
    end do:
    a ;
end proc:
for n from 0 to 13 do
    for k from 0 to n do
        printf("%d ",A079025(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Aug 28 2018
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
      b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 22 2019

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)]*b[n - i, Min[n - i, i]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)