A108461 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into pairs (i,j) with i,j>=1, not both 1.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 4, 2, 2, 1, 5, 2, 9, 2, 5, 1, 3, 2, 5, 4, 4, 5, 2, 3, 2, 7, 2, 11, 2, 11, 2, 7, 2, 2, 4, 7, 4, 5, 5, 4, 7, 4, 2, 1, 5, 4, 16, 2, 15, 2, 16, 4, 5, 1, 4, 2, 5, 9, 7, 5, 5, 7, 9, 5, 2, 4, 1, 11, 2, 11, 4, 21, 2, 21, 4, 11, 2, 11, 1, 2, 2, 11, 4, 5, 11, 7

Offset: 1

Author

Christian G. Bower, Jun 03 2005

Keywords

• nonn
• tabl

Comments

The rule of building products is (a,b)*(x,y) = (a*x,b*y).

The number of divisors of (n,k) is A143235(n,k)-1, where the subtraction of 1 means that the unit (1,1) is not admitted here. - R. J. Mathar, Nov 30 2017

Examples

			1 1 1 2 1 ...
1 2 2 4 2 ...
1 2 2 4 2 ...
2 4 4 9 4 ...
1 2 2 4 2 ...
(6,2)=(6,1)*(1,2)=(3,2)*(2,1)=(3,1)*(2,2)=(1,2)*(6,1), so a(6,2)=5.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1711, the first 59 diagonals.
• R. J. Mathar, Java Source code of 2 files to generate the b-file

Crossrefs

Cf. A051707, A108455-A108470.

Columns 1-3: A001055, A057567, A057567.

Main diagonal: A108462.

Formula

Dirichlet g.f.: A(s, t) = exp(B(s, t)/1 + B(2*s, 2*t)/2 + B(3*s, 3*t)/3 + ...) where B(s, t) = zeta(s)*zeta(t)-1.