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

1,5

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

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

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.

Example

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.

Crossrefs

Cf. A051707, A108455-A108470.

Columns 1-3: A001055, A057567, A057567.

Main diagonal: A108462.

Sequence in context: A286885 A365618 A254885 * A321004 A258595 A233544

Adjacent sequences: A108458 A108459 A108460 * A108462 A108463 A108464

Keyword

nonn,tabl

Author

Christian G. Bower, Jun 03 2005

Status

approved