A235992 Numbers with an even arithmetic derivative, cf. A003415.

0, 1, 4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 28, 32, 33, 35, 36, 39, 40, 44, 48, 49, 51, 52, 55, 56, 57, 60, 64, 65, 68, 69, 72, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 121, 123, 124, 128, 129, 132, 133

Offset: 1

Author

Reinhard Zumkeller, Mar 11 2014

Keywords

• nonn

Comments

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

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

Union of multiples of 4 and odd numbers with an even number of prime factors with multiplicity. - Charlie Neder, Feb 25 2019

After two initial terms (0 and 1), numbers n such that A086134(n) = 2. - Antti Karttunen, Sep 30 2019

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A235991 (complement).

Union of A327862 and A327864.

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

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

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

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

Programs

Haskell

a235992 n = a235992_list !! (n-1)
a235992_list = filter (even . a003415) [0..]

Mathematica

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

Python

from itertools import count, islice
from sympy import factorint
def A235992_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: not n&3 or (n&1 and not sum(factorint(n).values())&1), count(max(startvalue,0)))
A235992_list = list(islice(A235992_gen(),40)) # Chai Wah Wu, Nov 04 2022