A363063 - OEIS

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 #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