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A363063
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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.
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6
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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
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OFFSET
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1,2
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COMMENTS
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Includes all products of terms in A347284, but there are also other terms such as 4320.
Closed under multiplication. - Peter Munn, May 21 2023
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LINKS
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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).
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EXAMPLE
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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.
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)
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MATHEMATICA
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Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* Amiram Eldar, May 16 2023 *)
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PROG
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(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
return sorted(f(kmax, 0, kmax))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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