%I #45 May 12 2024 11:35:12
%S 1,2,2,4,2,5,2,6,4,5,2,10,2,5,5,9,2,10,2,10,5,5,2,16,4,5,6,10,2,14,2,
%T 12,5,5,5,20,2,5,5,16,2,14,2,10,10,5,2,24,4,10,5,10,2,16,5,16,5,5,2,
%U 28,2,5,10,16,5,14,2,10,5,14,2,32,2,5,10,10,5,14,2,24,9,5,2,28,5,5,5,16,2,28,5
%N Consider the products of all pairs of (not necessarily distinct) positive divisors of n. a(n) is the number of these products that divide n. a(n) also is the number of the products that are divisible by n.
%C Number of 3D grids of n congruent boxes with two different edge lengths, in a box, modulo rotation (cf. A034836 for cubes instead of boxes and A007425 for boxes with three different edge lengths; cf. A000005 for the 2D case). - _Manfred Boergens_, Feb 25 2021
%C Number of distinct faces obtainable by arranging n unit cubes into a cuboid. - _Chris W. Milson_, Mar 14 2021
%H Antti Karttunen, <a href="/A140773/b140773.txt">Table of n, a(n) for n = 1..10000</a>
%H Chris W. Milson, <a href="https://github.com/chrismilson/cuboid-areas">Constructing Cuboids</a>
%H Chris W. Milson, <a href="https://github.com/chrismilson/cuboid-areas/blob/e860c66ec6dfce910bafffe5ead15fa5548169cf/main.py#L49">A faster algorithm for a(n)</a>
%F a(n) = Sum_{m|n} A038548(m) = Sum_{m|n} ceiling(d(m)/2), where d(m) = number of divisors of m (A000005). - _Manfred Boergens_, Feb 25 2021
%F a(n) = Sum_{d|n} A135539(d,n/d). - _Ridouane Oudra_, Jul 10 2021
%F a(n) = (A007425(n) + A046951(n))/2. - _Ridouane Oudra_, Apr 10 2024
%e The divisors of 20 are 1,2,4,5,10,20. There are 10 pairs of divisors whose product divides 20: 1*1=1, 1*2=2, 1*4=4, 1*5=5, 1*10=10, 1*20=20, 2*2=4, 2*5=10, 2*10=20, 4*5 = 20. Likewise, there are 10 products that are divisible by 20: 4*5=20, 2*10=20, 4*10=40, 10*10=100, 1*20=20, 2*20=40, 4*20=80, 5*20=100, 10*20=200, 20*20=400. So a(20) = 10.
%t (* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Count[ n/Times @@@ Union[Sort /@ Tuples[Divisors@ n, 2]], _Integer]; Array[f, 91] (* _Robert G. Wilson v_, May 31 2008 *)
%t d=Divisors[n]; r=Length[d]; Sum[Ceiling[Length[Divisors[d[[j]]]]/2],{j,r}] (* _Manfred Boergens_, Feb 25 2021 *)
%o (PARI)
%o \\ Two implementations, after the two different interpretations given by the author of the sequence:
%o A140773v1(n) = { my(ds = divisors(n),s=0); for(i=1,#ds,for(j=i,#ds,if(!(n%(ds[i]*ds[j])),s=s+1))); s; }
%o A140773v2(n) = { my(ds = divisors(n),s=0); for(i=1,#ds,for(j=i,#ds,if(!((ds[i]*ds[j])%n),s=s+1))); s; }
%o \\ _Antti Karttunen_, May 19 2017
%o (Python) # See C. W. Milson link.
%Y Cf. A140774.
%Y Cf. A000005, A034836, A038548, A007425, A046951.
%Y Cf. A369255 (parity), A369256 (positions of odd terms).
%K nonn
%O 1,2
%A _Leroy Quet_, May 29 2008
%E Corrected and extended by _Robert G. Wilson v_, May 31 2008