A383178 Numbers k such that omega(k) = 4 and p^omega(k) < k^(1/4) < lpf(k)^(omega(k)+1) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

81719, 268801, 565471, 626603, 631997, 657169, 700321, 799459, 838457, 893513, 916453, 1108927, 1212083, 1239389, 1271209, 1354681, 1366817, 1408637, 1420763, 1500313, 1527619, 1574359, 1602137, 1639877, 1700557, 1719871, 1751173, 1758203, 1775341, 1783511, 1843969

Offset: 1

Author

Michael De Vlieger, May 09 2025

Keywords

• nonn

Comments

Let primes p, q, r, s, p < q < r < s, divide k = a(n).

Then floor(log(k)/log(p)) = 4 and floor(log(k)/log(q)) = floor(log(k)/log(r)) = floor(log(k)/log(s)) = 3.

Let R(k) = row k of A162306 = {m <= k : rad(m) | k}. Then A010846(k) = c(k) is the number of terms in row k of A162306.

A010846(a(n)) = 51 for k such that p*r^3 < k.

A010846(a(n)) = 50 for k such that p*r^3 > k.

Examples

			Table of n, a(n), prime decomposition of a(n), and A010846(n) = c(n) for n = 1..12:
 n      a(n)    facs(a(n))  c(n)  p*r^3
--------------------------------------------------
 1     81719   11*17*19*23   51   11*19^3 = 75449
 2    268801   13*23*29*31   50   13*29^3 = 317057
 3    565471   17*29*31*37   51   17*31^3 = 506447
 4    626603   17*29*31*41   51   17*31^3 = 506447
 5    631997   19*29*31*37   51   19*31^3 = 566029
 6    657169   17*29*31*43   51   17*31^3 = 506447
 7    700321   19*29*31*41   51   19*31^3 = 566029
 8    799459   17*31*37*41   50   17*37^3 = 861101
 9    838457   17*31*37*43   50   17*37^3 = 861101
10    893513   19*31*37*41   50   19*37^3 = 962407
11    916453   17*31*37*47   51   17*37^3 = 861101
12   1108927   17*37*41*43   50   17*41^3 = 1171657
Let S(k) = row k of A162306 = {m <= k : rad(m) | k}.
Writing p^a*q^b*r^c*s^d instead as "abcd" (i.e., catenating prime power exponents), the following combinations are in S(a(n)). In brackets we show p*r^3, which is in S(a(n)) for n such that c(n) = 51, but not in S(a(n)) for n such that c(n) = 50.
0000 1000 2000 3000 4000    0010 1010 2010 3010    0020 1020 2020    0030 [1030]
0100 1100 2100 3100         0110 1110 2110         0120 1120
0200 1200 2200              0210 1210
0300 1300
.
0001 1001 2001 3001         0011 1011 2011         0021
0101 1101 2101              0111 1111
0201 1201
.
0002 1002 2002              0012
0102
.
0003
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Hasse diagram of R(268801) with logarithmic vertical scale. Gray represents the empty product, red represents primes, gold represents proper prime powers, green squarefree composites, and blue numbers that are neither squarefree nor prime powers.
• Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 8X vertical exaggeration. The green bar at the bottom of the graph emphasizes the x axis that rides on the top edge of the bar.

Crossrefs

Cf. A010846, A020639, A162306, A383177.

Programs

Mathematica

f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
  If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
  Union@ Reap[Do[
    While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
      If[Or[k == 1, Union[#2] == #1 - 1 & @@
        TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
        Sow[k]];
      j = 1; w[[-j]]++];
      If[j == i, Break[], j++; w[[-j]]++;
        w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
f[4, 2000000]