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

%I A323730 #16 Jan 27 2019 08:46:47
%S A323730 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 A323730 4851,1,6348,1,13,25,325,1,19474560,1,1,976,1,17,1089,9537,18513,1,
%U A323730 1225,1,19,1,1521,70840000,107747640000,1,81,1,1,23,1,343000,3763008,245790720
%N A323730 Table read by rows: row n lists every number j whose n-th power has exactly j divisors.
%C A323730 Row n lists every j such that tau(j^n) = j.
%C A323730 Since tau(1^n) = tau(1) = 1 for all n, every row of the table includes 1 as a term.
%C A323730 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 A323730 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 A323730 Jon E. Schoenfield, <a href="/A323730/a323730.txt">Rows 0..100 of the table</a>
%H A323730 Jon E. Schoenfield, <a href="/A323730/a323730_2.txt">Magma program for computing rows 0..23 of the table</a>
%F A323730 A073049(n) = T(n,2) if row n contains more than 1 term, 0 otherwise.
%F A323730 A323731(n) is the number of terms in row n.
%F A323730 A323732 lists the numbers n such that row n contains only the single term 1.
%F A323730 A323733 lists the numbers n such that row n contains more than one term; i.e., A323733 is the complement of A323732.
%F A323730 A323734(n) = T(n, A323731(n)) is the largest term in row n.
%e A323730 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 A323730 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 A323730 Row n=5 includes no terms other than 1 because there exists no number j > 1 such that tau(j^5) = j.
%e A323730 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 A323730 Table begins as follows:
%e A323730    n | row n
%e A323730   ---+---------------------------------
%e A323730    0 | 1;
%e A323730    1 | 1, 2;
%e A323730    2 | 1, 3;
%e A323730    3 | 1, 28, 40;
%e A323730    4 | 1, 5, 9, 45;
%e A323730    5 | 1;
%e A323730    6 | 1, 7;
%e A323730    7 | 1, 225;
%e A323730    8 | 1, 153;
%e A323730    9 | 1, 640;
%e A323730   10 | 1, 11, 441, 2541, 4851;
%e A323730   11 | 1, 6348;
%e A323730   12 | 1, 13, 25, 325;
%e A323730   13 | 1, 19474560;
%e A323730   14 | 1;
%e A323730   15 | 1, 976;
%e A323730   16 | 1, 17, 1089, 9537, 18513;
%e A323730   17 | 1, 1225;
%e A323730   18 | 1, 19;
%e A323730   19 | 1, 1521, 70840000, 107747640000;
%e A323730   20 | 1, 81;
%e A323730   21 | 1;
%e A323730   22 | 1, 23;
%e A323730   23 | 1, 343000, 3763008, 245790720;
%Y A323730 Cf. A073049 (Least m > 1 such that m^n has m divisors, or 0 if no such m exists).
%Y A323730 Cf. A000005, A323731, A323732, A323733, A323734.
%K A323730 nonn,tabf
%O A323730 0,3
%A A323730 _Jon E. Schoenfield_, Jan 25 2019