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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.
13
36, 60, 72, 84, 90, 100, 108, 120, 126, 132, 140, 144, 150, 156, 168, 180, 196, 198, 200, 204, 210, 216, 220, 225, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 308, 312, 315, 324, 330, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 390
OFFSET
1,1
COMMENTS
Positions of nonzero terms in A322437 or A322438.
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
Numbers with more semiprime divisors than prime divisors. - Wesley Ivan Hurt, Jun 10 2021
LINKS
Christophe Cordero, Factorizations of Cyclic Groups and Bayonet Codes, arXiv:2301.13566 [math.CO], 2023, p. 20.
EXAMPLE
An example of such a pair for 36 is (4*9)|(6*6).
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], Select[Subsets[facs[#], {2}], And[!Or@@Divisible@@@Tuples[#], !Or@@Divisible@@@Reverse/@Tuples[#]]&]!={}&]
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_ndf_pair(fac1, fac2) = { for(i=1, #fac1, for(j=1, #fac2, if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]), return(0)))); (1); };
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); };
isA320632(n) = has_at_least_one_ndfpair(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020
CROSSREFS
The following are additional cross-references relating to Granvik's conjecture.
bigomega(n) * omega(n) is A113901(n).
(bigomega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - bigomega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Sequence in context: A341283 A260138 A260131 * A368832 A188633 A328961
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 09 2018
STATUS
approved