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

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, 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, 2, 2, 5, 4, 5, 2, 3, 3, 1, 4, 2, 5, 2, 2, 2, 2, 2, 2, 1

Offset: 0

Author

Jon E. Schoenfield, Jan 26 2019

Keywords

• nonn

Comments

a(n) is the number of terms in row n of A323730.

Since 1^n = 1 has exactly 1 divisor for all n, a(n) >= 1.

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.

A323733 lists the numbers j such that a(j) > 1 (i.e., such that A073049(j) > 0).

Examples

			a(0) = 1 because there is only one number k whose 0th power (k^0 = 1) has exactly k divisors (namely, k=1).
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.
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.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000005, A073049, A323730, A323732, A323733, A323734.