%I #35 Jan 21 2023 09:33:05

%S 0,1,4,8,9,12,15,16,20,21,24,25,28,32,33,35,36,39,40,44,48,49,51,52,

%T 55,56,57,60,64,65,68,69,72,76,77,80,81,84,85,87,88,91,92,93,95,96,

%U 100,104,108,111,112,115,116,119,120,121,123,124,128,129,132,133

%N Numbers with an even arithmetic derivative, cf. A003415.

%C A165560(a(n)) = 0; A003415(a(n)) mod 2 = 0.

%C For n > 1: A007814(a(n)) <> 1, A006519(a(n)) <> 2.

%C Union of multiples of 4 and odd numbers with an even number of prime factors with multiplicity. - _Charlie Neder_, Feb 25 2019

%C After two initial terms (0 and 1), numbers n such that A086134(n) = 2. - _Antti Karttunen_, Sep 30 2019

%C A multiplicative semigroup; if m and n are in the sequence then so is m*n. (See also comments in A359780.) - _Antti Karttunen_, Jan 17 2023

%H Reinhard Zumkeller, <a href="/A235992/b235992.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Range[0, 133], EvenQ@ If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]] &] (* _Michael De Vlieger_, Sep 30 2019 *)

%o (Haskell)

%o a235992 n = a235992_list !! (n-1)

%o a235992_list = filter (even . a003415) [0..]

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A235992_gen(startvalue=0): # generator of terms >= startvalue

%o return filter(lambda n: not n&3 or (n&1 and not sum(factorint(n).values())&1), count(max(startvalue,0)))

%o A235992_list = list(islice(A235992_gen(),40)) # _Chai Wah Wu_, Nov 04 2022

%Y Cf. A235991 (complement).

%Y Union of A327862 and A327864.

%Y Union of A359829 (primitive elements) and A359831 (nonprimitive elements).

%Y Cf. A003415, A086134, A327863, A327865, A327933, A327935, A358680 (characteristic function).

%Y Positions of multiples of 4 in A358669 (and in A358765).

%Y Cf. also A028260, A036349, A046337, A332820 (other multiplicative semigroups), and comments in A359780.

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, Mar 11 2014