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A365535
Composite numbers k such that the core and the kernel of k are equal.
1
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
OFFSET
1,1
COMMENTS
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
LINKS
FORMULA
Union of A097054, A120944, and A362594.
EXAMPLE
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.
MATHEMATICA
Select[Range[150], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 2]], OddQ] &] (* Amiram Eldar, Sep 08 2023 *)
PROG
(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))
A365535_list = list(islice(A365535_gen(), 30)) # Chai Wah Wu, Sep 15 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Michel Marcus, Sep 08 2023
STATUS
approved