A108461 -id:A108461 - cp's OEIS frontend

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).

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)).

A108465 Table read by antidiagonals: T(n,k) (n>=2) = number of factorizations of (n,k) into pairs (i,j) with i,j>1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 4, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 2

Christian G. Bower, Jun 03 2005

(a,b)*(x,y)=(a*x,b*y).

			1 1 1 1 1 ...
1 1 1 1 1 ...
1 1 2 1 2 ...
1 1 1 1 1 ...
1 1 2 1 3 ...
(8,6)=(4,3)*(2,2)=(4,2)*(2,3), so a(8,6)=3.

Cf. A108461. Columns 4, 6: A038548 (n>1), A032741. Main diagonal: A108466.

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)-1)*(zeta(t)-1).

cp's OEIS Frontend

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).

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

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A108465 Table read by antidiagonals: T(n,k) (n>=2) = number of factorizations of (n,k) into pairs (i,j) with i,j>1.

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