OFFSET
2,1
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.
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.
FORMULA
EXAMPLE
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).
CROSSREFS
Largest primitive nondeficient numbers meeting other criteria: A287581.
KEYWORD
nonn
AUTHOR
David A. Corneth and Peter Munn, Oct 26 2020
STATUS
approved