A235991 Numbers with an odd arithmetic derivative, cf. A003415.

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 27, 29, 30, 31, 34, 37, 38, 41, 42, 43, 45, 46, 47, 50, 53, 54, 58, 59, 61, 62, 63, 66, 67, 70, 71, 73, 74, 75, 78, 79, 82, 83, 86, 89, 90, 94, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 113

Offset: 1

Author

Reinhard Zumkeller, Mar 11 2014

Keywords

• nonn

Comments

A165560(a(n)) = 1; A003415(a(n)) mod 2 = 1;

A007814(a(n)) <= 1, A006519(a(n)) <= 2.

Links

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

Crossrefs

Cf. A003415, A006519, A007814, A165560, A235992 (complement), A000040 (subsequence).

Programs

Haskell

a235991 n = a235991_list !! (n-1)
a235991_list = filter (odd . a003415) [0..]

Mathematica

ader[n_] := ader[n] = Switch[n, 0|1, 0, _, If[PrimeQ[n], 1,
     Sum[Module[{p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
Select[Range[120], OddQ[ader[#]]&] (* Jean-François Alcover, Oct 10 2021 *)

Python

from itertools import count, islice
from sympy import factorint
def A235991_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: n&3==2 or (n&1 and sum(factorint(n).values())&1), count(max(startvalue,0)))
A235991_list = list(islice(A235991_gen(),40)) # Chai Wah Wu, Nov 04 2022

Formula

n is in this sequence iff either n is congruent to 2 modulo 4 or n and Omega(n) are both odd. - Charlie Neder, Feb 25 2019