A108462 -id:A108462 - cp's OEIS frontend

A108463 a(n) = A108462(A025487(n)).

1, 2, 9, 15, 31, 92, 109, 444, 203, 339, 712, 1903, 1663, 1043, 4070, 7279, 10105, 2998, 20061, 16601, 25904, 4140, 27036, 51551, 8406, 86713, 117750, 86172, 43263, 149570, 229964, 22652, 342999, 682147, 272252, 318223, 717296, 198091, 930988

Offset: 1

Christian G. Bower, Jun 03 2005

Number of factorizations of (k,k) into pairs (i,j) by prime signature.

Pairs (i,j) must satisfy i,j>=1, not both 1; (a,b)*(x,y) = (a*x,b*y).

Cf. A025487, A108462.

Offset changed to 1 and a(27)-a(29) from Jinyuan Wang, Jan 17 2022

a(30)-a(39) from Amiram Eldar, Mar 04 2023

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.

Original entry on oeis.org

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

Christian G. Bower, Jun 03 2005

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

			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.

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

Cf. A051707, A108455-A108470.
Columns 1-3: A001055, A057567, A057567.
Main diagonal: A108462.

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.

cp's OEIS Frontend

A108463 a(n) = A108462(A025487(n)).

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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.

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