A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 11520, 12288, 13824, 16384, 17280, 18432

Offset: 1

Author

Pontus von Brömssen, May 16 2023

Keywords

• nonn

Comments

Includes all products of terms in A347284, but there are also other terms such as 4320.

Closed under multiplication. - Peter Munn, May 21 2023

Examples

			151200 = 2^5 * 3^3 * 5^2 * 7 is a term, because 2^5 >= 3^3 >= 5^2 >= 7.
72 = 2^3 * 3^2 is not a term, because 2^3 < 3^2.
40 = 2^3 * 3^0 * 5 is not a term, because 3^0 < 5.
From _Michael De Vlieger_, May 19 2023: (Start)
Sequence read as an irregular triangle delimited by appearance of 2^m:
     1
     2
     4
     8     12
    16     24
    32     48
    64     96
   128    144    192
   256    288    384
   512    576    720    768    864
  1024   1152   1440   1536   1728
  2048   2304   2880   3072   3456
  4096   4320   4608   5760   6144   6912
  8192   8640   9216  10368  11520  12288  13824
  ... (End)
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n, pi(p)), n = 1..1024, showing multiplicity e with a color function such that e = 1 is black, e = 2 is red, e = 3 is orange, etc., 12X vertical exaggeration. On the bottom, a color code represents a(n) is empty product (black), prime (red), composite prime power (gold), neither squarefree nor prime power (blue).
• Michael De Vlieger, Plot multiplicities e in a(n) = Product p^e at (x,y) = (e, -n) for n = 1..1024, 8X horizontal exaggeration.

Crossrefs

Cf. A181818, A305720, A347284, A363098 (primitive terms).

Subsequence of: A087980, {1} U A363122.

Programs

Mathematica

Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* Amiram Eldar, May 16 2023 *)

Python

from sympy import nextprime
primes = [2] # global list of first primes
def f(kmax,pi,ppmax):
    # Generate numbers up to kmax with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    ppmax = min(ppmax,kmax)
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(kmax//pp,pi+1,pp):
            yield pp*x
        pp *= p0
def A363063_list(kmax):
    return sorted(f(kmax,0,kmax))