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A365535
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Composite numbers k such that the core and the kernel of k are equal.
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1
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6, 8, 10, 14, 15, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 51, 54, 55, 56, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 88, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 125, 128, 129, 130, 133, 134, 135, 136, 138
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OFFSET
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1,1
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COMMENTS
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In other words composite numbers k such that the squarefree part of k and the squarefree kernel of k are equal (A007913(k) = A007947(k)). The definition excludes 1 and primes because in those cases it is trivially true that the core and kernel are equal (to 1).
A composite number k is in this sequence iff all of its prime power factors have odd exponents. A072587 is the complement of this sequence within the composites, A002808.
Composite exponentially odd numbers. - Amiram Eldar, Sep 08 2023
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LINKS
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FORMULA
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EXAMPLE
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6, 10, 14, 15, 21,... are all terms because they are composite squarefree
8, 27, 32,... are all terms because they are all odd prime powers.
24 = 2^3*3^1 is a term because its prime power factors (1,3) are both odd.
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MATHEMATICA
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Select[Range[150], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 2]], OddQ] &] (* Amiram Eldar, Sep 08 2023 *)
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PROG
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(PARI) isok(k) = if (!isprime(k) && (k>1), core(k) == factorback(factorint(k)[, 1])); \\ Michel Marcus, Sep 08 2023
(Python)
from itertools import count, islice
from sympy import factorint
def A365535_gen(): # generator of terms
return (n for n in count(2) if sum(f:=factorint(n).values())>1 and all(d&1 for d in f))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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