

A338427


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


1



6, 20, 2205, 12705, 117234117, 42840834309, 2792098376579421, 674431969285588989475, 21526530767769616227341527825, 292210459765634328314801626540200511773, 292210459765634328314801626540200511773
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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), 413426.
Peter Munn, PARI program
Eric Weisstein's World of Mathematics, Smooth Number.
Index entries for sequences related to sigma(n)


FORMULA

a(n) = Max_{m <= n, k >= 1} A338133(m, k).
a(n) = max( {m in A006039 : A006530(m) <= A000040(n)} ).


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 31smooth primitive nondeficient (and primitive abundant) number, 292210459765634328314801626540200511773. So a(12) = a(11).


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 divisorrelated criteria: A211198, A273057.
Sequence in context: A216912 A175671 A222741 * A069257 A133885 A170867
Adjacent sequences: A338424 A338425 A338426 * A338428 A338429 A338430


KEYWORD

nonn


AUTHOR

David A. Corneth and Peter Munn, Oct 26 2020


STATUS

approved



