A108455 -id:A108455 - cp's OEIS frontend

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

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.

A051707 Number of factorizations of (n,n) into pairs (j,k).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 8, 3, 5, 1, 23, 1, 5, 5, 23, 1, 23, 1, 23, 5, 5, 1, 91, 3, 5, 8, 23, 1, 52, 1, 60, 5, 5, 5, 143, 1, 5, 5, 91, 1, 52, 1, 23, 23, 5, 1, 328, 3, 23, 5, 23, 1, 91, 5, 91, 5, 5, 1, 339, 1, 5, 23, 161, 5, 52, 1, 23, 5, 52, 1, 686, 1, 5, 23, 23, 5, 52, 1, 328, 23, 5, 1, 339, 5

Offset: 1

Yasutoshi Kohmoto

Pairs (j,k) must satisfy j>1, k>=1; (a,b)*(x,y)=(a*x,b*y); unit is (1,1).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

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

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A050354, A108461, A108455, A348161 (into at most two pairs).
a(A025487) = A108460.
a(p^k) = A108457(k).
a(A002110) = A108459.
Main diagonal of A108455.

Edited by Christian G. Bower, Jun 03 2005

A348157 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into one or two pairs (i,j) with i > 0, j > 0 (and if i=1 then j=1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 2, 0, 1, 1, 3, 1, 3, 1, 0, 1, 1, 2, 1, 3, 1, 2, 0, 1, 1, 3, 1, 4, 1, 3, 2, 0, 1, 1, 2, 1, 3, 1, 3, 2, 2, 0, 1, 1, 3, 1, 5, 1, 4, 2, 3, 1, 0, 1, 1, 3, 1, 3, 1, 3, 3, 3, 1, 3, 0, 1, 1, 3, 1, 5, 1, 5, 2, 4, 1, 5, 1, 0, 1, 1, 2, 1, 4, 1, 3, 3, 3, 1, 5, 1, 2

Offset: 1

Sean A. Irvine, Oct 03 2021

(a,b)*(x,y) = (a*x,b*y); unit is (1,1).

			Table begins
  1 0 0 0 0 ...
  1 1 1 1 1 ...
  1 1 1 1 1 ...
  2 2 2 3 2 ...
  1 1 1 1 1 ...
  ...
(6,4) = (3,4)*(2,1) = (3,1)*(2,4) = (3,2)*(2,2), so a(6,4)=4.

Cf. A108455 (any number of pairs), A108461. Column 1: A001055.

rows = 14; t[n_, m_] := Ceiling[(DivisorSigma[0, n] - 2)*DivisorSigma[0, m]/2]+1; t[1, 1] = 1; t[1, ] = 0; ft = Flatten[ Table[ t[n-m+1, m], {n, 1, rows}, {m, n, 1, -1}]] (* _Jean-François Alcover, Nov 18 2011, after Franklin T. Adams-Watters *)

For n > 1, T(n,m) = ceiling((tau(n)-2)*tau(m)/2) + 1, where tau(n) = A000005(n). - Franklin T. Adams-Watters, Jun 23 2010.

A108456 Table read by antidiagonals: T(n,k) = number of partitions of (n,k) into pairs (i,j) with i>0, j>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 1, 3, 6, 7, 7, 0, 1, 4, 8, 12, 12, 11, 0, 1, 4, 10, 16, 21, 19, 15, 0, 1, 5, 12, 23, 31, 36, 30, 22, 0, 1, 5, 15, 28, 45, 55, 58, 45, 30, 0, 1, 6, 17, 37, 60, 84, 94, 92, 67, 42, 0, 1, 6, 20, 44, 80, 115, 147, 153, 140, 97, 56, 0, 1, 7, 23

Offset: 0

Christian G. Bower, Jun 03 2005

(a,b)+(x,y)=(a+x,b+y); unit is (0,0).

			1 0 0 0 0 ...
1 1 1 1 1 ...
2 2 3 3 4 ...
3 4 6 8 10 ...
5 7 12 16 23 ...
(3,2)=(2,2)+(1,0)=(2,1)+(1,1)=(2,0)+(1,2)=(1,2)+(1,0)+(1,0)=(1,1)+(1,1)+(1,0), so a(3,2)=6.

N. J. A. Sloane, Transforms

Cf. A108461, A108455. Columns 0-1: A000041, A000070. Main diagonal: A108457.

Euler transform of table whose g.f. is x/((1-x)*(1-y)).

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A051707 Number of factorizations of (n,n) into pairs (j,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A348157 Table read by antidiagonals: T(n,k) = number of factorizations of (n,k) into one or two pairs (i,j) with i > 0, j > 0 (and if i=1 then j=1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A108456 Table read by antidiagonals: T(n,k) = number of partitions of (n,k) into pairs (i,j) with i>0, j>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula