A365535 Composite numbers k such that the core and the kernel of k are equal.

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

Author

David James Sycamore and Michael De Vlieger, Sep 08 2023

Keywords

• nonn

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

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A002808 and A268335.

Cf. A005117, A006881, A007913, A007947, A072587, A097054, A120944, A268335, A362594.

Programs

Mathematica

Select[Range[150], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 2]], OddQ] &] (* Amiram Eldar, Sep 08 2023 *)

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

Formula

Union of A097054, A120944, and A362594.

Extensions

More terms from Michel Marcus, Sep 08 2023