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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.

%I A381250 #17 May 06 2025 23:50:18
%S A381250 1,2,6,1001,81719,101007559,84248643949,78464111896111,
%T A381250 997804397813471821,1314665322768473913751,25030469300030639321689313,
%U A381250 93516019518175801382127421211,1873482639168918364977596279806547,60958708904928776821774364389940352443,1089851191947047137351117158610882538395561
%N 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.
%C A381250 Terms are squarefree.
%e A381250 Let lpf = A020639, slpf = A119288, and gpf = A006530.
%e A381250 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.
%e A381250                                            prime indices
%e A381250  n                                  a(n)   lpf  slpf-gpf  prime factors
%e A381250 -------------------------------------------------------------------------
%e A381250  0                                    1     0             -
%e A381250  1                                    2     1             2
%e A381250  2                                    6     1     2       2*3
%e A381250  3                                 1001     4     5-6     7*11*13
%e A381250  4                                81719     5     7-9     11*17*19*23
%e A381250  5                            101007559     9    13-16    23*41*43*47*53
%e A381250  6                          84248643949    12    19-23    etc.
%e A381250  7                       78464111896111    17    25-30
%e A381250  8                   997804397813471821    26    41-47
%e A381250  9               1314665322768473913751    32    48-55
%e A381250 10           25030469300030639321689313    47    69-77
%e A381250 11        93516019518175801382127421211    56    83-92
%e A381250 12   1873482639168918364977596279806547    73   108-118
%e A381250 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.
%e A381250 a(0) = 1 since 1 is the only number that does not have prime factors.
%e A381250 a(1) = 2 since prime numbers have just 1 prime factor, and 2 is the smallest prime.
%e A381250 a(2) = 6 since f(2,6) = 2 and f(3,6) = 1; 6 is the smallest squarefree semiprime.
%e A381250 a(3) = 1001 since w(1001) = {3,2,2} and is the smallest sphenic number with this property.
%e A381250 30 is not in the sequence since w(30) = {4,3,2}; 42 is not in since w(42) = {5,3,1}, etc.
%e A381250 a(4) = 81719 since w(81719) = {4,3,3,3} and is the smallest number with 4 distinct prime factors with this property, etc.
%t A381250 f[om_, lm_] := Block[{f, i, j, k, nn, p, q, w, z},
%t A381250   i = Abs[om]; z = i - 1; j = z; nn = Abs[lm]; w = ConstantArray[1, i];
%t A381250   Catch@ Do[
%t A381250     While[Set[{k, p, q}, {Times @@ #, #[[1]], #[[2]]}] &@
%t A381250       Map[Prime, Accumulate@ w]; k <= nn,
%t A381250       If[And[q^i > k, p^(i + 1) > k], Throw[k]];
%t A381250     j = z; w[[-j]]++];
%t A381250     If[j == i, Break[], j++; w[[-j]]++;
%t A381250     w = PadRight[w[[;; -j]], i, 1]], {ii, Infinity}] ];
%t A381250 {1, 2}~Join~Table[f[n, 2^(11*n + 2)], {n, 2, 16}]
%Y A381250 Cf. A001221, A005117, A006530, A020639, A119288, A138109, A382022.
%K A381250 nonn
%O A381250 0,2
%A A381250 _Michael De Vlieger_, Apr 21 2025