A381250 a(n) = least k with n distinct prime factors such that floor(log_q(k)) = floor(log_p(k))-1, where p is the smallest prime factor of k, and q is any other distinct prime factor of k.

1, 2, 6, 1001, 81719, 101007559, 84248643949, 78464111896111, 997804397813471821, 1314665322768473913751, 25030469300030639321689313, 93516019518175801382127421211, 1873482639168918364977596279806547, 60958708904928776821774364389940352443, 1089851191947047137351117158610882538395561

Offset: 0

Author

Michael De Vlieger, Apr 21 2025

Keywords

• nonn

Comments

Terms are squarefree.

Examples

			Let lpf = A020639, slpf = A119288, and gpf = A006530.
Table of a(n), n=0..12, listing the indices of the smallest, second smallest, and greatest prime factors, the latter 2 pertaining to n >= 2 and n >= 3, respectively.
                                           prime indices
 n                                  a(n)   lpf  slpf-gpf  prime factors
-------------------------------------------------------------------------
 0                                    1     0             -
 1                                    2     1             2
 2                                    6     1     2       2*3
 3                                 1001     4     5-6     7*11*13
 4                                81719     5     7-9     11*17*19*23
 5                            101007559     9    13-16    23*41*43*47*53
 6                          84248643949    12    19-23    etc.
 7                       78464111896111    17    25-30
 8                   997804397813471821    26    41-47
 9               1314665322768473913751    32    48-55
10           25030469300030639321689313    47    69-77
11        93516019518175801382127421211    56    83-92
12   1873482639168918364977596279806547    73   108-118
Let f(p,k) = floor(log_p k) and let w be the list of f(p,k) across the sorted list of distinct prime factors of k.
a(0) = 1 since 1 is the only number that does not have prime factors.
a(1) = 2 since prime numbers have just 1 prime factor, and 2 is the smallest prime.
a(2) = 6 since f(2,6) = 2 and f(3,6) = 1; 6 is the smallest squarefree semiprime.
a(3) = 1001 since w(1001) = {3,2,2} and is the smallest sphenic number with this property.
30 is not in the sequence since w(30) = {4,3,2}; 42 is not in since w(42) = {5,3,1}, etc.
a(4) = 81719 since w(81719) = {4,3,3,3} and is the smallest number with 4 distinct prime factors with this property, etc.
		

Crossrefs

Cf. A001221, A005117, A006530, A020639, A119288, A138109, A382022.

Programs

Mathematica

f[om_, lm_] := Block[{f, i, j, k, nn, p, q, w, z},
  i = Abs[om]; z = i - 1; j = z; nn = Abs[lm]; w = ConstantArray[1, i];
  Catch@ Do[
    While[Set[{k, p, q}, {Times @@ #, #[[1]], #[[2]]}] &@
      Map[Prime, Accumulate@ w]; k <= nn,
      If[And[q^i > k, p^(i + 1) > k], Throw[k]];
    j = z; w[[-j]]++];
    If[j == i, Break[], j++; w[[-j]]++;
    w = PadRight[w[[;; -j]], i, 1]], {ii, Infinity}] ];
{1, 2}~Join~Table[f[n, 2^(11*n + 2)], {n, 2, 16}]