A382926 Irregular table where row n lists numbers k in row n of A162306 for which there exists a prime p | n such that k*p > n.

2, 3, 4, 5, 3, 4, 6, 7, 8, 9, 4, 5, 8, 10, 11, 6, 8, 9, 12, 13, 4, 7, 8, 14, 5, 9, 15, 16, 17, 8, 9, 12, 16, 18, 19, 5, 8, 10, 16, 20, 7, 9, 21, 4, 8, 11, 16, 22, 23, 9, 12, 16, 18, 24, 25, 4, 8, 13, 16, 26, 27, 7, 8, 14, 16, 28, 29, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30

Offset: 2

Author

Michael De Vlieger, Apr 28 2025

Keywords

• nonn
• tabf

Comments

The number n appears in each row. For n in A024619, for all p|n, p^floor(log_p n) is in row n. Thus, the number of terms in row n for n in A024619 is at least 1+omega(n), where omega = A001221 is the number of distinct prime factors of n.

Examples

			Let s(n) = A382964(n).
Table of select rows:
 n  s(n)    row n of this sequence
--------------------------------------------------------
 6    3     3,  4,  6;
10    4     4,  5,  8, 10;
12    4     6,  8,  9, 12;
14    4     4,  7,  8, 14;
15    3     5,  9, 15;
18    5     8,  9, 12, 16, 18;
20    5     5,  8, 10, 16, 20;
21    3     7,  9, 21;
22    5     4,  8, 11, 16, 22;
24    5     9, 12, 16, 18, 24;
26    5     4,  8, 13, 16, 26;
28    5     7,  8, 14, 16, 28;
30   12     8,  9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30.
In the examples below, we place terms in row n in brackets [] among other terms in row n of A162306, presented in order of row n of A275280.
Row p^m for m > 0 and prime p is {p^m}, since multiplying p^m by p exceeds p^m.
Row 10 = {4, 5, 8, 10}, since numbers k such that rad(k) | 10 contains these numbers, furthermore, we have the following: 2 or 5 times 8 exceeds 10, 5*4 > 10, 2 or 5 times 10 exceeds 10, and 5*5 > 10.
      1    2   [4]  [8]
     [5] [10]
Row 24 = {9, 12, 16, 18, 24}, since numbers k such that rad(k) | 24 contains these numbers, furthermore, we have the following: 2 or 3 times 16 exceeds 24, 2 or 3 times 24 exceeds 24, 3*12 > 24, 2 or 3 times 18 exceeds 24, and 3*9 > 24.
      1    2    4    8  [16]
      3    6  [12] [24]
     [9] [18]
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..11938 (rows n = 2..1000, flattened)

Crossrefs

Cf. A000961, A007947, A024619, A162306, A275280, A382964 (row lengths).

Programs

Mathematica

(* First, run the "regs" function from A369609, then: *)
Table[Select[regs[n], Function[k, AnyTrue[FactorInteger[n][[All, 1]], #*k > n &]]], {n, 2, 30}] // Flatten

Formula

For n in A000961, row n is {n}.