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

1,10

Comments

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

Links

Table of n, a(n) for n=1..105.

Formula

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.

Example

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A227003 A307431 A263848 * A108455 A193759 A117468

Adjacent sequences: A348154 A348155 A348156 * A348158 A348159 A348160

Keyword

nonn,tabl

Author

Sean A. Irvine, Oct 03 2021

Status

approved