A394860 Numbers that are neither squarefree nor squareful, with exponents of prime power factors that are distinct.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 75, 76, 80, 88, 92, 96, 98, 99, 104, 112, 116, 117, 124, 135, 136, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 207, 208, 212, 224, 232, 236, 242, 244, 245, 248, 250, 261

Offset

1,1

Comments

Intersection of A130091 and A332785.

{a(n)} = A332785 / A392926.

Smallest term with k distinct prime factors is A006939(k). Therefore, the set of composite superprimorials, A006939 \ {1, 2}, is a subset of this sequence.

Agrees with A345381 for 77 terms; a(78) = 2^3 * 3^2 * 5 is not of the form p^k * q, primes p != q, k > 1, thus, not in A345381.

Agrees with A348097 for 237 terms; A348097(238) = 1260 = 2^2 * 3^2 * 5 * 7 is not in this sequence.

Links

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

Example

Let s = A332785.
Table of n, a(n) for select n:
     n                a(n)
  ----------------------------------------------
     1       s(1) =    12 = 2^2 * 3
     2       s(2) =    18 = 2   * 3^2
     3       s(3) =    20 = 2^2 * 5
     4       s(4) =    24 = 2^3 * 3
     5       s(5) =    28 = 2^2 * 7
     6       s(6) =    40 = 2^3 * 5
     7       s(7) =    44 = 2^2 * 11
     8       s(8) =    45 = 3^2 * 5
    14      s(15) =    63 = 3^2 * 7
    78     s(112) =   360 = 2^3 * 3^2 * 5
   238     s(412) =  1200 = 2^4 * 3   * 5^2
  9249   s(29110) = 75600 = 2^4 * 3^3 * 5^2 * 7

Mathematica

fQ[x_] := And[MemberQ[#, 1], AnyTrue[#, # > 1 &], DuplicateFreeQ[#] ] &[FactorInteger[x][[All, -1]] ]; Select[Range[300], fQ]

Crossrefs

Cf. A006939, A130091, A332785, A345381, A348097, A392926.

Sequence in context: A177425 A182854 A348097 * A345381 A328930 A392537

Adjacent sequences: A394857 A394858 A394859 * A394861 A394862 A394863

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 14 2026

Status

approved