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

0,2

Comments

Terms are squarefree.

Links

Table of n, a(n) for n=0..14.

Example

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.

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}]

Crossrefs

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

Sequence in context: A113569 A252739 A178773 * A384000 A046857 A129454

Adjacent sequences: A381247 A381248 A381249 * A381251 A381252 A381253

Keyword

nonn

Author

Michael De Vlieger, Apr 21 2025

Status

approved