cp's OEIS Frontend

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

%I A365535 #25 Sep 15 2023 10:20:58
%S A365535 6,8,10,14,15,21,22,24,26,27,30,32,33,34,35,38,39,40,42,46,51,54,55,
%T A365535 56,57,58,62,65,66,69,70,74,77,78,82,85,86,87,88,91,93,94,95,96,102,
%U A365535 104,105,106,110,111,114,115,118,119,120,122,123,125,128,129,130,133,134,135,136,138
%N A365535 Composite numbers k such that the core and the kernel of k are equal.
%C A365535 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).
%C A365535 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.
%C A365535 Composite exponentially odd numbers. - _Amiram Eldar_, Sep 08 2023
%H A365535 Michael De Vlieger, <a href="/A365535/b365535.txt">Table of n, a(n) for n = 1..10000</a>
%F A365535 Union of A097054, A120944, and A362594.
%e A365535 6, 10, 14, 15, 21,... are all terms because they are composite squarefree
%e A365535 8, 27, 32,... are all terms because they are all odd prime powers.
%e A365535 24 = 2^3*3^1 is a term because its prime power factors (1,3) are both odd.
%t A365535 Select[Range[150], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 2]], OddQ] &] (* _Amiram Eldar_, Sep 08 2023 *)
%o A365535 (PARI) isok(k) = if (!isprime(k) && (k>1), core(k) == factorback(factorint(k)[, 1])); \\ _Michel Marcus_, Sep 08 2023
%o A365535 (Python)
%o A365535 from itertools import count, islice
%o A365535 from sympy import factorint
%o A365535 def A365535_gen(): # generator of terms
%o A365535     return (n for n in count(2) if sum(f:=factorint(n).values())>1 and all(d&1 for d in f))
%o A365535 A365535_list = list(islice(A365535_gen(),30)) # _Chai Wah Wu_, Sep 15 2023
%Y A365535 Intersection of A002808 and A268335.
%Y A365535 Cf. A005117, A006881, A007913, A007947, A072587, A097054, A120944, A268335, A362594.
%K A365535 nonn
%O A365535 1,1
%A A365535 _David James Sycamore_ and _Michael De Vlieger_, Sep 08 2023
%E A365535 More terms from _Michel Marcus_, Sep 08 2023