A393644 Squarefree numbers k with m distinct prime factors, such that gpf(k) < lpf(k)^m, where gpf = A006530 and lpf = A020639.

6, 15, 21, 30, 35, 42, 55, 65, 70, 77, 85, 91, 95, 105, 115, 119, 133, 143, 161, 165, 187, 195, 203, 209, 210, 217, 221, 231, 247, 253, 255, 259, 273, 285, 287, 299, 301, 319, 323, 329, 330, 341, 345, 357, 377, 385, 390, 391, 399, 403, 407, 429, 437, 451, 455

Offset

1,1

Comments

Proper subset of A120944, superset of A138109.

Let omega = A001221 and pi = A000720.

Smallest a(n) with omega(a(n)) = m is a(n) = A002110(m), since prime(m) < 2^m for m > 1.

There are a finite number of k with 1 < omega(k) = m such that p | k for any given least prime factor p. For k with omega(k) = 2 (i.e., for k in A138109), there are pi(prime(i)^2) - i = A079047(i) numbers k with lpf(k) = prime(i).

This sequence includes A384000(n) for n > 1.

Links

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

Example

Table of n, a(n) for select n:
      n      a(n)
  ---------------------------------------------
      1        6 = 2 * 3
      2       15 = 3 * 5
      3       21 = 3 * 7
      4       30 = 2 * 3 * 5
      5       35 = 5 * 7
      6       42 = 2 * 3 * 7
     25      210 = 2 * 3 * 5 * 7
    114     1001 = 7 * 11 * 13
    253     2310 = 2 * 3 * 5 * 7 * 11
   3281    30030 = 2 * 3 * 5 * 7 * 11 * 13
  27220   268801 = 13 * 23 * 29 * 31
  51003   510510 = 2 * 3 * 5 * 7 * 11 * 13 * 17

Mathematica

s = Select[Range[500], 1 < PrimeNu[#] == PrimeOmega[#] &]; Select[s, Function[{p, q, k}, q < p^k] @@ {First[#], Last[#], Length[#]} &[FactorInteger[#][[;; , 1]] ] &]

Crossrefs

Cf. A001221, A002110, A005117, A006530, A020639, A120944, A138109, A384000.

Sequence in context: A009092 A330205 A015793 * A373899 A261078 A063466

Adjacent sequences: A393640 A393641 A393642 * A393645 A393646 A393647

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 18 2026

Status

approved