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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.

%I A363063 #41 May 27 2025 14:57:45
%S A363063 1,2,4,8,12,16,24,32,48,64,96,128,144,192,256,288,384,512,576,720,768,
%T A363063 864,1024,1152,1440,1536,1728,2048,2304,2880,3072,3456,4096,4320,4608,
%U A363063 5760,6144,6912,8192,8640,9216,10368,11520,12288,13824,16384,17280,18432
%N 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.
%C A363063 Includes all products of terms in A347284, but there are also other terms such as 4320.
%C A363063 Closed under multiplication. - _Peter Munn_, May 21 2023
%H A363063 Pontus von Brömssen, <a href="/A363063/b363063.txt">Table of n, a(n) for n = 1..10000</a>
%H A363063 Michael De Vlieger, <a href="/A363063/a363063.png">Plot p^e | a(n) at (x,y) = (n, pi(p))</a>, 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).
%H A363063 Michael De Vlieger, <a href="/A363063/a363063_1.png">Plot multiplicities e in a(n) = Product p^e at (x,y) = (e, -n)</a> for n = 1..1024, 8X horizontal exaggeration.
%e A363063 151200 = 2^5 * 3^3 * 5^2 * 7 is a term, because 2^5 >= 3^3 >= 5^2 >= 7.
%e A363063 72 = 2^3 * 3^2 is not a term, because 2^3 < 3^2.
%e A363063 40 = 2^3 * 3^0 * 5 is not a term, because 3^0 < 5.
%e A363063 From _Michael De Vlieger_, May 19 2023: (Start)
%e A363063 Sequence read as an irregular triangle delimited by appearance of 2^m:
%e A363063      1
%e A363063      2
%e A363063      4
%e A363063      8     12
%e A363063     16     24
%e A363063     32     48
%e A363063     64     96
%e A363063    128    144    192
%e A363063    256    288    384
%e A363063    512    576    720    768    864
%e A363063   1024   1152   1440   1536   1728
%e A363063   2048   2304   2880   3072   3456
%e A363063   4096   4320   4608   5760   6144   6912
%e A363063   8192   8640   9216  10368  11520  12288  13824
%e A363063   ... (End)
%t A363063 Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* _Amiram Eldar_, May 16 2023 *)
%o A363063 (Python)
%o A363063 from sympy import nextprime
%o A363063 primes = [2] # global list of first primes
%o A363063 def f(kmax,pi,ppmax):
%o A363063     # Generate numbers up to kmax with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
%o A363063     if len(primes) <= pi: primes.append(nextprime(primes[-1]))
%o A363063     p0 = primes[pi]
%o A363063     ppmax = min(ppmax,kmax)
%o A363063     if ppmax < p0:
%o A363063         yield 1
%o A363063         return
%o A363063     pp = 1
%o A363063     while pp <= ppmax:
%o A363063         for x in f(kmax//pp,pi+1,pp):
%o A363063             yield pp*x
%o A363063         pp *= p0
%o A363063 def A363063_list(kmax):
%o A363063     return sorted(f(kmax,0,kmax))
%Y A363063 Cf. A181818, A305720, A347284, A363098 (primitive terms).
%Y A363063 Subsequence of: A087980, {1} U A363122.
%K A363063 nonn
%O A363063 1,2
%A A363063 _Pontus von Brömssen_, May 16 2023