A323730 Table read by rows: row n lists every number j whose n-th power has exactly j divisors.

1, 1, 2, 1, 3, 1, 28, 40, 1, 5, 9, 45, 1, 1, 7, 1, 225, 1, 153, 1, 640, 1, 11, 441, 2541, 4851, 1, 6348, 1, 13, 25, 325, 1, 19474560, 1, 1, 976, 1, 17, 1089, 9537, 18513, 1, 1225, 1, 19, 1, 1521, 70840000, 107747640000, 1, 81, 1, 1, 23, 1, 343000, 3763008, 245790720

Offset: 0

Author

Jon E. Schoenfield, Jan 25 2019

Keywords

• nonn
• tabf

Comments

Row n lists every j such that tau(j^n) = j.

Since tau(1^n) = tau(1) = 1 for all n, every row of the table includes 1 as a term.

Each prime p appears as a term in row p-1 since, for n=p-1, tau(p^n) = tau(p^(p-1)) = p.

Examples

			Row n=3 includes 28 as a term because tau(28^3) = tau((2^2 * 7)^3) = tau(2^6 * 7^3) = (6+1)*(3+1) = 7*4 = 28.
Row n=3 includes 40 as a term because tau(40^3) = tau((2^3 * 5)^3) = tau(2^9 * 5^3) = (9+1)*(3+1) = 10*4 = 40.
Row n=5 includes no terms other than 1 because there exists no number j > 1 such that tau(j^5) = j.
Row n=23 includes 245790720 as a term because tau(245790720^23) = tau((2^11 * 3^3 * 5 * 7 * 127)^23) = tau(2^253 * 3^69 * 5^23 * 7^23 * 127^23) = (253+1)*(69+1)(23+1)*(23+1)*(23+1) = 254*70*24^3 = 245790720.
Table begins as follows:
   n | row n
  ---+---------------------------------
   0 | 1;
   1 | 1, 2;
   2 | 1, 3;
   3 | 1, 28, 40;
   4 | 1, 5, 9, 45;
   5 | 1;
   6 | 1, 7;
   7 | 1, 225;
   8 | 1, 153;
   9 | 1, 640;
  10 | 1, 11, 441, 2541, 4851;
  11 | 1, 6348;
  12 | 1, 13, 25, 325;
  13 | 1, 19474560;
  14 | 1;
  15 | 1, 976;
  16 | 1, 17, 1089, 9537, 18513;
  17 | 1, 1225;
  18 | 1, 19;
  19 | 1, 1521, 70840000, 107747640000;
  20 | 1, 81;
  21 | 1;
  22 | 1, 23;
  23 | 1, 343000, 3763008, 245790720;
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 0..282 (all terms of rows 0..100)
• Jon E. Schoenfield, Rows 0..100 of the table
• Jon E. Schoenfield, Magma program for computing rows 0..23 of the table

Crossrefs

Cf. A073049 (Least m > 1 such that m^n has m divisors, or 0 if no such m exists).

Cf. A000005, A323731, A323732, A323733, A323734.

Formula

A073049(n) = T(n,2) if row n contains more than 1 term, 0 otherwise.

A323731(n) is the number of terms in row n.

A323732 lists the numbers n such that row n contains only the single term 1.

A323733 lists the numbers n such that row n contains more than one term; i.e., A323733 is the complement of A323732.

A323734(n) = T(n, A323731(n)) is the largest term in row n.