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%I #36 May 21 2023 12:56:01
%S 1,2,4,8,12,16,24,32,48,64,96,128,144,192,256,288,384,512,576,720,768,
%T 864,1024,1152,1440,1536,1728,2048,2304,2880,3072,3456,4096,4320,4608,
%U 5760,6144,6912,8192,8640,9216,10368,11520,12288,13824,16384,17280,18432
%N 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 Includes all products of terms in A347284, but there are also other terms such as 4320.
%C Closed under multiplication. - _Peter Munn_, May 21 2023
%H Pontus von Brömssen, <a href="/A363063/b363063.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 151200 = 2^5 * 3^3 * 5^2 * 7 is a term, because 2^5 >= 3^3 >= 5^2 >= 7.
%e 72 = 2^3 * 3^2 is not a term, because 2^3 < 3^2.
%e 40 = 2^3 * 3^0 * 5 is not a term, because 3^0 < 5.
%e From _Michael De Vlieger_, May 19 2023: (Start)
%e Sequence read as an irregular triangle delimited by appearance of 2^m:
%e 1
%e 2
%e 4
%e 8 12
%e 16 24
%e 32 48
%e 64 96
%e 128 144 192
%e 256 288 384
%e 512 576 720 768 864
%e 1024 1152 1440 1536 1728
%e 2048 2304 2880 3072 3456
%e 4096 4320 4608 5760 6144 6912
%e 8192 8640 9216 10368 11520 12288 13824
%e ... (End)
%t Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* _Amiram Eldar_, May 16 2023 *)
%o (Python)
%o from sympy import nextprime
%o primes = [2] # global list of first primes
%o def f(kmax,pi,ppmax):
%o # Generate numbers up to kmax with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
%o if len(primes) <= pi: primes.append(nextprime(primes[-1]))
%o p0 = primes[pi]
%o ppmax = min(ppmax,kmax)
%o if ppmax < p0:
%o yield 1
%o return
%o pp = 1
%o while pp <= ppmax:
%o for x in f(kmax//pp,pi+1,pp):
%o yield pp*x
%o pp *= p0
%o def A363063_list(kmax):
%o return sorted(f(kmax,0,kmax))
%Y Cf. A087980, A181818, A305720, A347284, A363098 (primitive terms).
%K nonn
%O 1,2
%A _Pontus von Brömssen_, May 16 2023
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