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

1,3

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

Mathematica

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

Prog

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

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.

Sequence in context: A145190 A327907 A177713 * A362006 A359783 A359829

Adjacent sequences: A235989 A235990 A235991 * A235993 A235994 A235995

Keyword

nonn

Author

Reinhard Zumkeller, Mar 11 2014

Status

approved