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A359178
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Numbers with a unique smallest prime exponent.
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33
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2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117
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OFFSET
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1,1
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COMMENTS
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180 is the smallest number with a unique smallest prime exponent that is not a member of A130091.
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LINKS
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EXAMPLE
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2 = 2^1 is a term since it has 1 as a unique smallest exponent.
6 = 2^1 * 3^1 is not a term since it has two primes with the same smallest exponent.
180 = 2^2 * 3^2 * 5^1 is a term since it has 1 as a unique smallest exponent.
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MATHEMATICA
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q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Count[e, Min[e]] == 1]; Select[Range[2, 200], q] (* Amiram Eldar, Jan 08 2023 *)
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PROG
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(Python)
from sympy import factorint
def ok(k):
c = sorted(factorint(k).values())
return len(c) == 1 or c[0] != c[1]
print([k for k in range(2, 118) if ok(k)])
(Python)
from itertools import count, islice
from sympy import factorint
def A359178_gen(startvalue=2): # generator of terms >= startvalue
return filter(lambda n:(f:=list(factorint(n).values())).count(min(f))==1, count(max(startvalue, 2)))
(PARI) isok(n) = if (n>1, my(f=factor(n), e = vecmin(f[, 2])); #select(x->(x==e), f[, 2], 1) == 1); \\ Michel Marcus, Jan 27 2023
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CROSSREFS
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For parts instead of multiplicities we have A247180, counted by A002865.
Partitions of this type are counted by A362610.
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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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