A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

50, 75, 80, 98, 112, 135, 147, 189, 240, 242, 245, 252, 270, 294, 300, 336, 338, 350, 352, 360, 363, 378, 396, 416, 450, 468, 480, 490, 504, 507, 525, 528, 540, 550, 560, 578, 588, 594, 600, 605, 612, 624, 650, 672, 684, 700, 702, 720, 722, 726, 735, 750, 756

Offset

1,1

Comments

Does not contain 3-smooth numbers.

Contains neither A168263 nor A367511.

Conjecture: contains most highly composite numbers.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Union of {k = m*s : rad(m) | s, max(p, q) <= m < s}, where s is in A120944.

{a(n)} = A364702 \ A366250.

{a(n)} = A361098 \ A341645.

Example

Let q = A053669(k) and let p = A119288(k).
For s = 10, we have {50, 80}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 10*{ max(5, 3) <= m < 10 : rad(m) | 10 }
   = 10*{5, 8} = {50, 80}.
For s = 15, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 15*{ max(5, 2) <= m < 15 : rad(m) | 15 }
   = 15*{5, 9} = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.
For s = 30, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 30*{ max(3, 7) <= m < 30 : rad(m) | 30 }
   = 30*{8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27}
   = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.

Mathematica

nn = 756;
Select[Select[Range[12, nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &],
  And[Max[#2, #3] <= #1 < #4, ! AllTrue[#5, # > 1 &]] & @@
    {#1/#4, #2, #3, #4, #5} & @@
    {#1, #2[[2, 1]], #3, Times @@ #2[[All, 1]], #2[[All, -1]]} & @@
    {#, FactorInteger[#], If[OddQ[#], 2,
        q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

Crossrefs

Cf. A002182, A003586, A053669, A119288, A120944, A168263, A341645, A361098, A364702, A366250, A367511.

Sequence in context: A039473 A309123 A146170 * A217857 A071366 A321517

Adjacent sequences: A367705 A367706 A367707 * A367709 A367710 A367711

Keyword

nonn

Author

Michael De Vlieger, Feb 09 2024

Status

approved