A338427 a(n) is the largest prime(n)-smooth primitive nondeficient number.

6, 20, 2205, 12705, 117234117, 42840834309, 2792098376579421, 674431969285588989475, 21526530767769616227341527825, 292210459765634328314801626540200511773, 292210459765634328314801626540200511773

Offset: 2

Author

David A. Corneth and Peter Munn, Oct 26 2020

Keywords

• nonn

Comments

See A006039 for a definition and list of primitive nondeficient numbers.

The first prime being 2, the prime(1)-smooth numbers are the powers of 2, which are all deficient. So a(1) is undefined, and the sequence offset is 2.

Omitting the initial "6" gives us the largest prime(n)-smooth primitive abundant numbers (based on their A071395 definition). Using the variant definition of primitive abundant from A091191, the equivalent sequence starts 18, 30, 2205, 12705, 117234117, ... .

If m is a prime(n)-smooth primitive nondeficient number, the odd part of m divides a member of one of the first (n - 1) finite sets described in the Dickson reference and the even part of m is less than 2^A035100(n). This provides an upper bound for such numbers, meaning there is a largest prime(n)-smooth primitive nondeficient number for all n >= 2.

Examples

			Initial terms, showing factorization:
   n          a(n)
   2             6 = 2 * 3,
   3            20 = 2^2 * 5,
   4          2205 = 3^2 * 5 * 7^2,
   5         12705 = 3 * 5 * 7 * 11^2,
   6     117234117 = 3^2 * 7^2 * 11^2 * 13^3,
   7   42840834309 = 3^4 * 7^2 * 13^3 * 17^3,
   ...
The largest primitive nondeficient (and primitive abundant) number that has prime(12) = 37 as largest prime factor is 29504726357465429322218597476548828125, which is one digit shorter than the largest 31-smooth primitive nondeficient (and primitive abundant) number, 292210459765634328314801626540200511773. So a(12) = a(11).
		

Links

• Peter Munn, Table of n, a(n) for n = 2..18
• L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
• Peter Munn, PARI program
• Eric Weisstein's World of Mathematics, Smooth Number.
• Index entries for sequences related to sigma(n)

Crossrefs

After removing duplicate terms we get a subsequence of A006039, A338133.

Cf. A000040, A006530, A035100, A071395, A091191.

The largest prime(n)-smooth numbers meeting other divisor-related criteria: A211198, A273057.

Largest primitive nondeficient numbers meeting other criteria: A287581.

Formula

a(n) = Max_{m <= n, k >= 1} A338133(m, k).

a(n) = max( {m in A006039 : A006530(m) <= A000040(n)} ).