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
Michael De Vlieger, Table of n, a(n) for n = 1..10000
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))
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore and Michael De Vlieger, Sep 08 2023
EXTENSIONS
More terms from Michel Marcus, Sep 08 2023
STATUS
approved