A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4

Offset

1,120

Comments

First differs from A322437 at a(144) = 4, A322437(144) = 3.

First differs from A379958 at a(120) = 2, A379958(120) = 1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n.

Example

The a(240) = 5 pairs of factorizations::
  (4*4*15)|(4*6*10)
    (6*40)|(15*16)
    (8*30)|(12*20)
   (10*24)|(15*16)
   (12*20)|(15*16)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
divpropQ[x_, y_]:=And[x!=y, Divisible[x, y]];
Table[Length[Select[Subsets[facs[n], {2}], And[!Or@@divpropQ@@@Tuples[#], !Or@@divpropQ@@@Reverse/@Tuples[#]]&]], {n, 100}]

Prog

(PARI)
factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z, Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf, d); z = factorizations(n/d, d, newf, z))); (z));
is_proper_ndf_pair(fac1, fac2) = { for(i=1, #fac1, for(j=1, #fac2, if((fac1[i]!=fac2[j]) && (!(fac1[i]%fac2[j]) || !(fac2[j]%fac1[i])), return(0)))); (1); };
number_of_proper_ndfpairs(z) = sum(i=1, #z, sum(j=i+1, #z, is_proper_ndf_pair(z[i], z[j])));
A322438(n) = number_of_proper_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Jan 24 2025

Crossrefs

Cf. A001055, A285572, A285573, A303362, A303386, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322437, A322442, A379958.

Sequence in context: A317576 A070205 A138363 * A080225 A122841 A326075

Adjacent sequences: A322435 A322436 A322437 * A322439 A322440 A322441

Keyword

nonn,changed

Author

Gus Wiseman, Dec 08 2018

Extensions

Data section extended up to a(144) by Antti Karttunen, Jan 24 2025

Status

approved