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

15,1

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).

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.

Example

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

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]]

Crossrefs

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

Sequence in context: A283080 A329652 A077493 * A290663 A290544 A290830

Adjacent sequences: A384957 A384958 A384959 * A384961 A384962 A384963

Keyword

nonn

Author

Michael De Vlieger, Jul 06 2025

Status

approved