A291045 Numbers with prime factorization such that the cube of a lesser prime in the factorization is greater than the square of a greater prime in the factorization.

15, 30, 35, 45, 55, 60, 70, 75, 77, 90, 91, 105, 110, 119, 120, 135, 140, 143, 150, 154, 165, 175, 180, 182, 187, 195, 209, 210, 220, 221, 225, 231, 238, 240, 245, 247, 253, 255, 270, 273, 275, 280, 285, 286, 299, 300, 308, 315, 319, 323, 330, 341, 345, 350, 357, 360, 364, 374, 375, 377

Offset: 1

Author

Richard Locke Peterson, Aug 16 2017

Keywords

• nonn

Comments

Definition rephrased: if n is a number with prime divisors p and q with p < q but p^3 > q^2, then n will be in the sequence, otherwise, not.

Sequence is a superclosed semigroup; that is, if s is in the sequence and x is any number, then x*s is in the sequence: if s in the sequence, there are primes p,q dividing s with p < q, p^3 > q^2, so p and q would also divide x*s.

Examples

			6 = 2*3 is not in the sequence since 2^3 < 3^2.
15 = 3*5 is in the sequence because 3^3 > 5^2.
		

Crossrefs

Cf. A289484.

Programs

Maple

isA291045 := proc(n)
    local pdivs,i,j;
    pdivs := sort(convert(numtheory[factorset](n),list)) ;
    for i from 1 to nops(pdivs)-1 do
    for j from i+1 to nops(pdivs) do
        if op(i,pdivs)^3 > op(j,pdivs)^2 then
            return true;
        end if;
    end do:
    end do:
    false;
end proc:
for n from 1 to 400 do
    if isA291045(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 04 2017

Mathematica

Select[Range@ 400, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^3 > #2^2 & @@ # &] &] (* Michael De Vlieger, Aug 17 2017 *)