A273057 -id:A273057 - cp's OEIS frontend

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

David A. Corneth and Peter Munn, Oct 26 2020

nonn

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.

			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).

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)

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.

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

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

A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.

Original entry on oeis.org

3, 10, 17, 28, 27, 43, 44, 69, 68, 58, 97, 97, 125, 164, 201, 185, 162, 254, 263, 313, 491, 434, 466, 417, 309, 358, 510, 633, 935, 1148, 454

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 18 2016

Theorem. The sequence is unbounded.
Proof. Since the sequence of highly composite numbers (A002182) is a subsequence of this sequence, it is sufficient to prove that the number M_n of highly composite numbers with the maximal prime divisor p_n is unbounded. Let N be a large highly composite number. Then for the greatest prime divisor p_N of N we have [Erdos] p_N=O(log N). So for all N<=x, p_N=O(log x).
If M_n=O(1), then the number of all highly composite numbers <=x is O(p_n)=O(log x). However, Erdos [Erdos] proved that this number is more than (log x)^(1+c) for a certain c>0.
So we have a contradiction. This means that M_n and this sequence are unbound. QED

P. Erdős, On Highly composite numbers, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.

Cf. A067128, A273015, A273016, A273018, A273057.

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

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