%I #29 Jan 17 2023 21:25:08

%S 6,20,2205,12705,117234117,42840834309,2792098376579421,

%T 674431969285588989475,21526530767769616227341527825,

%U 292210459765634328314801626540200511773,292210459765634328314801626540200511773

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

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

%C 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.

%C 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, ... .

%C 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.

%H Peter Munn, <a href="/A338427/b338427.txt">Table of n, a(n) for n = 2..18</a>

%H L. E. Dickson, <a href="http://www.jstor.org/stable/2370405">Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors</a>, Amer. J. Math., 35 (1913), 413-426.

%H Peter Munn, <a href="/A338427/a338427.txt">PARI program</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

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

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

%e Initial terms, showing factorization:

%e n a(n)

%e 2 6 = 2 * 3,

%e 3 20 = 2^2 * 5,

%e 4 2205 = 3^2 * 5 * 7^2,

%e 5 12705 = 3 * 5 * 7 * 11^2,

%e 6 117234117 = 3^2 * 7^2 * 11^2 * 13^3,

%e 7 42840834309 = 3^4 * 7^2 * 13^3 * 17^3,

%e ...

%e 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).

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

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

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

%Y Largest primitive nondeficient numbers meeting other criteria: A287581.

%K nonn

%O 2,1

%A _David A. Corneth_ and _Peter Munn_, Oct 26 2020