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.

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

Antti Karttunen, Table of n, a(n) for n = 1..23437

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

Cf. A001055, A050336, A285572, A303362, A305149, A305193, A317144, A322435, A322437, A322439, A322440, A322441, A322442.

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).

Cf. A060687, A070175, A124010, A323023, A328956, A328957, A328960, A328961, A328963.

Sequence in context: A341283 A260138 A260131 * A368832 A188633 A328961

Adjacent sequences: A320629 A320630 A320631 * A320633 A320634 A320635

Keyword

nonn

Author

Gus Wiseman, Dec 09 2018

Status

approved