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

1,1

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A345297 A336533 A359782 * A327906 A333866 A325362

Adjacent sequences: A235988 A235989 A235990 * A235992 A235993 A235994

Keyword

nonn

Author

Reinhard Zumkeller, Mar 11 2014

Status

approved