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Table read by rows: row n lists every number j whose n-th power has exactly j divisors.
6

%I #16 Jan 27 2019 08:46:47

%S 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,

%T 4851,1,6348,1,13,25,325,1,19474560,1,1,976,1,17,1089,9537,18513,1,

%U 1225,1,19,1,1521,70840000,107747640000,1,81,1,1,23,1,343000,3763008,245790720

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

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

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

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

%H Jon E. Schoenfield, <a href="/A323730/b323730.txt">Table of n, a(n) for n = 0..282</a> (all terms of rows 0..100)

%H Jon E. Schoenfield, <a href="/A323730/a323730.txt">Rows 0..100 of the table</a>

%H Jon E. Schoenfield, <a href="/A323730/a323730_2.txt">Magma program for computing rows 0..23 of the table</a>

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

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

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

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

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

%e 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.

%e 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.

%e Row n=5 includes no terms other than 1 because there exists no number j > 1 such that tau(j^5) = j.

%e 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.

%e Table begins as follows:

%e n | row n

%e ---+---------------------------------

%e 0 | 1;

%e 1 | 1, 2;

%e 2 | 1, 3;

%e 3 | 1, 28, 40;

%e 4 | 1, 5, 9, 45;

%e 5 | 1;

%e 6 | 1, 7;

%e 7 | 1, 225;

%e 8 | 1, 153;

%e 9 | 1, 640;

%e 10 | 1, 11, 441, 2541, 4851;

%e 11 | 1, 6348;

%e 12 | 1, 13, 25, 325;

%e 13 | 1, 19474560;

%e 14 | 1;

%e 15 | 1, 976;

%e 16 | 1, 17, 1089, 9537, 18513;

%e 17 | 1, 1225;

%e 18 | 1, 19;

%e 19 | 1, 1521, 70840000, 107747640000;

%e 20 | 1, 81;

%e 21 | 1;

%e 22 | 1, 23;

%e 23 | 1, 343000, 3763008, 245790720;

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

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

%K nonn,tabf

%O 0,3

%A _Jon E. Schoenfield_, Jan 25 2019