This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A322437 #31 May 28 2025 08:55:02
%S A322437 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 A322437 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 A322437 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
%N A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.
%C A322437 First differs from A322438 at a(144) = 3, A322438(144) = 4.
%C A322437 From _Antti Karttunen_, Dec 11 2020: (Start)
%C A322437 Zeros occur on numbers that are either of the form p^k, or q * p^k, or p*q*r, for some primes p, q, r, and exponent k >= 0. [Note also that in all these cases, when x > 1, A307408(x) = 2+A307409(x) = 2 + (A001222(x) - 1)*A001221(x) = A000005(x)].
%C A322437 Proof:
%C A322437 It is easy to see that for such numbers it is not possible to obtain two such distinct factorizations, that no factor of the other would not divide some factor of the other.
%C A322437 Conversely, the complement set of above is formed of such composites n that have at least one unitary divisor that is either of the form
%C A322437 (1) p^x * q^y, with x, y >= 2,
%C A322437 or
%C A322437 (2) p^x * q^y * r^z, with x >= 2, and y, z >= 1,
%C A322437 or
%C A322437 (3) p^x * q^y * r^z * s^w, with x, y, z, w >= 1,
%C A322437 where p, q, r, s are distinct primes. Let's indicate with C the remaining portion of k coprime to p, q, r and s (which could be also 1). Then in case (1) we can construct two factorizations, the first having factors (p*q*C) and (p^(x-1) * q^(y-1)), and the second having factors (p^x * C) and (q^y) that are guaranteed to satisfy the condition that no factor in the other factorization divides any of the factors of the other factorization. For case (2) pairs like {(p * q^y * C), (p^(x-1) * r^z)} and {(p^x * C), (q^y * r^z)}, and for case (3) pairs like {(p^x * q^y * C), (r^z * s^w)} and {(p^x * r^z * C), (q^y * s^w)} offer similar examples, therefore a(n) > 0 for all such cases.
%C A322437 (End)
%H A322437 Antti Karttunen, <a href="/A322437/b322437.txt">Table of n, a(n) for n = 1..65537</a>
%H A322437 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A322437 For n > 0, a(A002110(n)) = A322441(n)/2 = A339626(n). - _Antti Karttunen_, Dec 10 2020
%e A322437 The a(120) = 2 pairs of such factorizations:
%e A322437 (6*20)|(8*15)
%e A322437 (8*15)|(10*12)
%e A322437 The a(144) = 3 pairs of factorizations:
%e A322437 (6*24)|(9,16)
%e A322437 (8*18)|(12*12)
%e A322437 (9*16)|(12*12)
%e A322437 The a(210) = 3 pairs of factorizations:
%e A322437 (6*35)|(10*21)
%e A322437 (6*35)|(14*15)
%e A322437 (10*21)|(14*15)
%e A322437 [Note that 210 is the first squarefree number obtaining nonzero value]
%e A322437 The a(240) = 4 pairs of factorizations:
%e A322437 (6*40)|(15*16)
%e A322437 (8*30)|(12*20)
%e A322437 (10*24)|(15*16)
%e A322437 (12*20)|(15*16)
%e A322437 The a(1728) = 14 pairs of factorizations:
%e A322437 (6*6*48)|(27*64)
%e A322437 (6*12*24)|(27*64)
%e A322437 (6*288)|(27*64)
%e A322437 (8*8*27)|(12*12*12)
%e A322437 (12*12*12)|(27*64)
%e A322437 (12*12*12)|(32*54)
%e A322437 (12*144)|(27*64)
%e A322437 (12*144)|(32*54)
%e A322437 (16*108)|(24*72)
%e A322437 (18*96)|(27*64)
%e A322437 (24*72)|(27*64)
%e A322437 (24*72)|(32*54)
%e A322437 (27*64)|(36*48)
%e A322437 (32*54)|(36*48)
%t A322437 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A322437 Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]],{n,100}]
%o A322437 (PARI)
%o A322437 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 A322437 is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
%o A322437 number_of_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_ndf_pair(z[i],z[j])));
%o A322437 A322437(n) = number_of_ndfpairs(Vec(factorizations(n))); \\ _Antti Karttunen_, Dec 10 2020
%Y A322437 Cf. A000688, A001055, A285572, A285573, A294068, A303362, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322441, A322442, A339626.
%Y A322437 Cf. also comments A307408 and A307409.
%K A322437 nonn
%O A322437 1,120
%A A322437 _Gus Wiseman_, Dec 08 2018
%E A322437 Data section extended up to a(120) and more examples added by _Antti Karttunen_, Dec 10 2020