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

1,1

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

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

Intersection of A002808 and A268335.

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

Sequence in context: A130763 A332269 A328338 * A120497 A036436 A036455

Adjacent sequences: A365532 A365533 A365534 * A365536 A365537 A365538

Keyword

nonn

Author

David James Sycamore and Michael De Vlieger, Sep 08 2023

Extensions

More terms from Michel Marcus, Sep 08 2023

Status

approved