A317748 Irregular triangle where T(n,d) is the number of factorizations of n into factors > 1 with GCD d.

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1

Offset

1,18

Links

Table of n, a(n) for n=1..87.

Example

Triangle begins:
   1:  0
   2:  0  1
   3:  0  1
   4:  0  1  1
   5:  0  1
   6:  1  0  0  1
   7:  0  1
   8:  0  2  0  1
   9:  0  1  1
  10:  1  0  0  1
  11:  0  1
  12:  2  1  0  0  0  1
  13:  0  1
  14:  1  0  0  1
  15:  1  0  0  1
  16:  0  3  1  0  1
  17:  0  1
  18:  2  0  1  0  0  1
  19:  0  1
  20:  2  1  0  0  0  1

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
goc[n_, m_]:=Length[Select[facs[n], And[And@@(Divisible[#, m]&/@#), GCD@@(#/m)==1]&]];
Table[goc[n, d], {n, 30}, {d, Divisors[n]}]

Crossrefs

Row lengths are A000005. Row sums are A001055. First column is A281116. Number of nonzero terms in each row is A317751.

Cf. A007562, A007716, A014963, A045778, A050370, A162247, A289509, A303386, A303546.

Cf. A317752, A317755, A317757.

Sequence in context: A246720 A343030 A246690 * A090465 A364357 A052344

Adjacent sequences: A317745 A317746 A317747 * A317749 A317750 A317751

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 06 2018

Status

approved