A384960 a(n) = smallest sphenic number k such that A010846(k) = n.

1001, 105, 231, 30, 42, 70, 110, 66, 78, 170, 102, 114, 138, 370, 174, 826, 222, 246, 258, 318, 354, 402, 438, 498, 534, 582, 654, 762, 786, 894, 978, 1038, 1158, 1338, 1506, 1542, 1758, 1986, 2082, 2202, 2334, 2598, 2922, 3126, 3462, 3918, 4098, 4398, 4614, 5262

Offset: 15

Author

Michael De Vlieger, Jul 06 2025

Keywords

• nonn

Comments

a(1) = A384000(3) = 1001; A010846(1001) = A024718(3) = 15; 1001 is the smallest number k with 3 distinct prime factors that has the smallest possible number of terms in row k of A162306, i.e., m <= k such that rad(m) | k.

For n > 30, 6 | a(n).

Examples

			Table of a(n) indicating prime factors and S, where S = {ceiling(log_p a(n))} for all primes p that divide a(n), in order of the magnitude of p.
                                Prime power factor
                                    1111223344455
 n  m=a(n) pi(facs(m))    S     23571379391713739
-------------------------------------------------
15   1001   4.5.6       4.3.3   ...111
16    105   2.3.4       5.3.3   .111
17    231   2.4.5       5.3.3   .1.11
18     30   1.2.3       5.4.3   111
19     42   1.2.4       6.4.2   11.1
20     70   1.3.4       7.3.3   1.11
21    110   1.3.5       7.3.2   1.1.1
22     66   1.2.5       7.4.2   11..1
23     78   1.2.6       7.4.2   11...1
24    170   1.3.7       8.4.2   1.1...1
25    102   1.2.7       7.5.2   11....1
26    114   1.2.8       7.5.2   11.....1
27    138   1.2.9       8.5.2   11......1
28    370   1.3.12      9.4.2   1.1........1
29    174   1.2.10      8.5.2   11.......1
30    826   1.4.17     10.4.2   1..1............1
31    222   1.2.12      8.5.2   11.........1
32    246   1.2.13      8.6.2   11..........1
33    258   1.2.14      9.6.2   11...........1
34    318   1.2.16      9.6.2   11.............1
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 15..300
• Michael De Vlieger, Plot of terms k = p^a*q^b*r^c, primes p < q < r, in row a(n) of A162306, n = 15..50, at (x,y,z) = (a,b,c). For a(n) there are n blocks in each diagram.
• Michael De Vlieger, Mathematica code.

Crossrefs

Cf. A007304, A007947, A010846, A024718, A162306, A384000.

Programs

Mathematica

(* See Mathematica code link for function definitions for kappaomega and theta *)
s =  kappaomega[3, 6000]; t = Map[theta, s];
Map[s[[FirstPosition[t, #][[1]] ]] &, Union[t]]