A323730 -id:A323730 - cp's OEIS frontend

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

2, 3, 28, 5, 0, 7, 225, 153, 640, 11, 6348, 13, 19474560, 0, 976, 17, 1225, 19, 1521, 81, 0, 23, 343000, 49, 2601, 2133, 3025, 29, 1495296000, 31, 20063232, 4225, 15262600, 4761, 19236456, 37, 25462407801600, 5929, 34633600, 41, 0, 43, 7569, 356445, 8281

Offset: 1

Lekraj Beedassy, Aug 31 2002

Since a prime power p^k has exactly k+1 divisors, choosing k=p-1, where p is a prime, thus implies a(p-1)=p for any prime p.

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Cf. A000005, A323730.

Definition and terms for n = 1, 13, 23, 29, 31, 33, 35, 37, 39, and 44 corrected by Jon E. Schoenfield, Apr 17 2010

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

Original entry on oeis.org

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

Jon E. Schoenfield, Jan 26 2019

nonn

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).

			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.

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

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

A323734 a(n) is the largest number j whose n-th power has exactly j divisors.

Original entry on oeis.org

1, 2, 3, 40, 45, 1, 7, 225, 153, 640, 4851, 6348, 325, 19474560, 1, 976, 18513, 1225, 19, 107747640000, 81, 1, 23, 245790720, 49, 2601, 2133, 3025, 94221, 56241820800, 31, 20063232, 4225, 15262600, 4761, 19236456, 37, 25462407801600, 5929, 2952832000, 21921921

Offset: 0

Jon E. Schoenfield, Jan 26 2019

nonn

a(n) is the largest (and last) of the A323731(n) numbers in row n of A323730.

If a(n)=1 then n is a term in A323732; otherwise, n is a term in A323733.

			The numbers j whose 3rd powers have exactly j divisors are 1, 28, and 40; the largest of these is 40, so a(3) = 40.
The only number j whose 5th power has exactly j divisors is 1, so a(1) = 1.

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

Cf. A000005, A073049, A323730, A323731, A323732, A323733.

A323732 Numbers k for which there exists no j > 1 such that j^k has exactly j divisors.

Original entry on oeis.org

5, 14, 21, 41, 50, 54, 67, 76, 86, 90, 111, 113, 119, 131, 142, 153, 165, 175, 186, 202, 204, 216, 224, 230

Offset: 1

Jon E. Schoenfield, Jan 26 2019

This sequence lists the numbers k such that A073049(k) = 0.
Equivalently:
numbers k for which the only number j such that j^k has exactly j divisors is 1;
numbers k such that A323731(k)=1;
numbers k such that A323734(k)=1.
The complement of this sequence is A323733.
The next terms after a(24)=230 appear to be 233, 253, 269, 273, 285, 293, 303, 307, 318, 321, 328, 345, 354, 357, 369, 370, 373, 384, 393, 402, 410, 412, 414, 426, 429, 431, 440, 441, 445, 468, ...

			There exists no j > 1 such that j^5 has exactly j divisors, so 5 is a term.
For k=15 and j=976, j^k = 976^15 = (2^4 * 61)^15 = 2^60 * 61^15, which has exactly (60+1)*(15+1) = 61*16 = 976 = j divisors, so k=15 is not a term.

Cf. A000005, A073049, A323730, A323731, A323733, A323734.

A323733 Numbers k for which there exists at least one number j > 1 such that j^k has exactly j divisors.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73

Offset: 1

Jon E. Schoenfield, Jan 26 2019

nonn

Complement of A323732.
This sequence lists the numbers k such that A073049(k) > 0.
Equivalently:
numbers k for which 1 is not the only number j such that j^k has exactly j divisors;
numbers k such that A323731(k) > 1;
numbers k such that A323734(k) > 1.

			For k=9 and j=640, j^k = 640^9 = (2^7 * 5)^9 = 2^63 * 5^9, which has exactly (63+1)*(9+1) = 64*10 = 640 = j divisors, so k=9 is a term.
There exists no j > 1 such that j^14 has exactly j divisors, so 14 is not a term.

Cf. A000005, A073049, A323730, A323731, A323732, A323734.

cp's OEIS Frontend

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

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

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A323734 a(n) is the largest number j whose n-th power has exactly j divisors.

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A323732 Numbers k for which there exists no j > 1 such that j^k has exactly j divisors.

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A323733 Numbers k for which there exists at least one number j > 1 such that j^k has exactly j divisors.

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