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

1,1

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.

Links

Table of n, a(n) for n=1..60.

Example

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.

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 *)

Crossrefs

Cf. A289484.

Sequence in context: A051967 A162592 A239247 * A079877 A215951 A060700

Adjacent sequences: A291042 A291043 A291044 * A291046 A291047 A291048

Keyword

nonn

Author

Richard Locke Peterson, Aug 16 2017

Status

approved