A384000 - OEIS

A384000

Smallest number k with n distinct prime factors such that A010846(k) = A024718(n) (a tight lower bound), or -1 if such k does not exist.

1, 2, 6, 1001, 268801, 3433936673, 2603508937756211

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OFFSET

0,2

COMMENTS

These numbers k have the smallest A010846(k) for a number with n distinct prime factors.

a(7) <= 44024279725172939536207 = 677 * 1721 * 2011 * 2069 * 2083 * 2087 * 2089.

a(8) <= 32829974457045619959776094471833047127947.

LINKS

Table of n, a(n) for n=0..6.

EXAMPLE

Table of a(n), n = 0..6, showing prime decomposition and cardinality of row a(n) of A162306, c(n) = A010846(a(n)) = A024718(n).

n a(n) c(n) prime factors of a(n) a(n)

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0 1 1 -

1 2 2 2 A000040(1)

2 6 5 2, 3 A138109(1)

3 1001 15 7, 11, 13 A383177(1)

4 268801 50 13, 23, 29, 31 A383178(2)

5 3433936673 176 41, 83, 97, 101, 103 A383179(209)

6 2603508937756211 638 163, 373, 439, 457, 461, 463

Tables of terms m in r(a(n)) = row a(n) of A162306, writing instead only exponents i of prime power factors p^i | m for each p | a(n), written in order of the prime base:

For n = 2, i.e., squarefree semiprime k in A138109 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-2 table:

00 10 20

01 11

Thus row 6 of A162306 has the following elements:

1 2 4

3 6

For n = 3, i.e., sphenic k in A383177 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-3 table:

000 100 200 300 001 101 201 002

010 110 210 011 111

020 120

Thus row 1001 of A162306 has the following elements:

1 7 49 343 13 91 637 169

11 77 539 141 1001