cp's OEIS Frontend

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

%I A322438 #16 Jan 24 2025 18:37:23
%S A322438 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,
%T A322438 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,
%U A322438 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
%N A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.
%C A322438 First differs from A322437 at a(144) = 4, A322437(144) = 3.
%C A322438 First differs from A379958 at a(120) = 2, A379958(120) = 1.
%H A322438 Antti Karttunen, <a href="/A322438/b322438.txt">Table of n, a(n) for n = 1..65537</a>
%H A322438 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%e A322438 The a(240) = 5 pairs of factorizations::
%e A322438   (4*4*15)|(4*6*10)
%e A322438     (6*40)|(15*16)
%e A322438     (8*30)|(12*20)
%e A322438    (10*24)|(15*16)
%e A322438    (12*20)|(15*16)
%t A322438 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A322438 divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
%t A322438 Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@divpropQ@@@Tuples[#],!Or@@divpropQ@@@Reverse/@Tuples[#]]&]],{n,100}]
%o A322438 (PARI)
%o A322438 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));
%o A322438 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); };
%o A322438 number_of_proper_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_proper_ndf_pair(z[i],z[j])));
%o A322438 A322438(n) = number_of_proper_ndfpairs(Vec(factorizations(n))); \\ _Antti Karttunen_, Jan 24 2025
%Y A322438 Cf. A001055, A285572, A285573, A303362, A303386, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322437, A322442, A379958.
%K A322438 nonn
%O A322438 1,120
%A A322438 _Gus Wiseman_, Dec 08 2018
%E A322438 Data section extended up to a(144) by _Antti Karttunen_, Jan 24 2025