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A323731 a(n) is the number of numbers k whose n-th power has exactly k divisors.

%I A323731 #14 Sep 26 2019 04:19:02
%S A323731 1,2,2,3,4,1,2,2,2,2,5,2,4,2,1,2,5,2,2,4,2,1,2,4,2,2,2,2,5,4,2,2,2,2,
%T A323731 2,2,2,2,2,4,3,1,5,10,3,2,5,2,2,2,1,4,2,3,1,6,2,2,2,6,4,4,3,4,2,2,5,1,
%U A323731 2,2,5,4,5,2,3,3,1,4,2,5,2,2,2,2,2,2,1
%N A323731 a(n) is the number of numbers k whose n-th power has exactly k divisors.
%C A323731 a(n) is the number of terms in row n of A323730.
%C A323731 Since 1^n = 1 has exactly 1 divisor for all n, a(n) >= 1.
%C A323731 A323732 lists the numbers j such that a(j) = 1 (i.e., such that A073049(j) = 0); for each such j, the only number k whose j-th power has exactly k divisors is 1.
%C A323731 A323733 lists the numbers j such that a(j) > 1 (i.e., such that A073049(j) > 0).
%H A323731 Jon E. Schoenfield, <a href="/A323731/b323731.txt">Table of n, a(n) for n = 0..100</a>
%e A323731 a(0) = 1 because there is only one number k whose 0th power (k^0 = 1) has exactly k divisors (namely, k=1).
%e A323731 a(2) = 2 because there are two numbers k such that tau(k^2) = k: tau(1^2) = tau(1) = 1 and tau(3^2) = tau(9) = 3.
%e A323731 a(43) = 10 because there are 10 numbers k such that tau(k^43) = k: 1, 7569, 2197000, 4296680960, 11128700700, 16629093000, 223705109760, 19462344549120, 32521578186240, and 5580197619796800.
%Y A323731 Cf. A000005, A073049, A323730, A323732, A323733, A323734.
%K A323731 nonn
%O A323731 0,2
%A A323731 _Jon E. Schoenfield_, Jan 26 2019