cp's OEIS Frontend

A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.

%I A320632 #59 Apr 12 2023 11:29:59
%S A320632 36,60,72,84,90,100,108,120,126,132,140,144,150,156,168,180,196,198,
%T A320632 200,204,210,216,220,225,228,234,240,252,260,264,270,276,280,288,294,
%U A320632 300,306,308,312,315,324,330,336,340,342,348,350,360,364,372,378,380,390
%N A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.
%C A320632 Positions of nonzero terms in A322437 or A322438.
%C A320632 _Mats Granvik_ has conjectured that these are all the positive integers k such that sigma_0(k) - 2 > (bigomega(k) - 1) * omega(k), where sigma_0 = A000005, omega = A001221, and bigomega = A001222. - _Gus Wiseman_, Nov 12 2019
%C A320632 Numbers with more semiprime divisors than prime divisors. - _Wesley Ivan Hurt_, Jun 10 2021
%H A320632 Antti Karttunen, <a href="/A320632/b320632.txt">Table of n, a(n) for n = 1..23437</a>
%H A320632 Christophe Cordero, <a href="https://arxiv.org/abs/2301.13566">Factorizations of Cyclic Groups and Bayonet Codes</a>, arXiv:2301.13566 [math.CO], 2023, p. 20.
%e A320632 An example of such a pair for 36 is (4*9)|(6*6).
%t A320632 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A320632 Select[Range[100],Select[Subsets[facs[#],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]!={}&]
%o A320632 (PARI)
%o A320632 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 A320632 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 A320632 has_at_least_one_ndfpair(z) = { for(i=1,#z,for(j=i+1,#z,if(is_ndf_pair(z[i],z[j]),return(1)))); (0); };
%o A320632 isA320632(n) = has_at_least_one_ndfpair(Vec(factorizations(n))); \\ _Antti Karttunen_, Dec 10 2020
%Y A320632 Cf. A001055, A050336, A285572, A303362, A305149, A305193, A317144, A322435, A322437, A322439, A322440, A322441, A322442.
%Y A320632 The following are additional cross-references relating to Granvik's conjecture.
%Y A320632 bigomega(n) * omega(n) is A113901(n).
%Y A320632 (bigomega(n) - 1) * omega(n) is A307409(n).
%Y A320632 sigma_0(n) - bigomega(n) * omega(n) is A328958(n).
%Y A320632 sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
%Y A320632 Cf. A060687, A070175, A124010, A323023, A328956, A328957, A328960, A328961, A328963.
%K A320632 nonn
%O A320632 1,1
%A A320632 _Gus Wiseman_, Dec 09 2018