A165560 The arithmetic derivative of n, modulo 2.

0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset

0,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for characteristic functions

Formula

a(n) = A003415(n) mod 2.

a(n) = (1-(-1)^n')/2.

a(A235991(n)) = 1 and a(A235992(n)) = 0. - Reinhard Zumkeller, Mar 11 2014

a(n) = 1 - A358680(n) = A358680(n) - A359792(n) = A358771(n) + A358773(n). - Antti Karttunen, Jan 16 2023

Maple

with(numtheory);
P:=proc(i)
local f, n, p, pfs;
for n from 0 by 1 to i do
  pfs:=ifactors(n)[2]; f:=n*add(op(2, p)/op(1, p), p=pfs);
  print(1/2*(1-(-1)^f));
od;
end:
P(1000);

Mathematica

d[0] = d[1] = 0; d[n_] := n*Total[f = FactorInteger[n]; f[[All, 2]]/f[[All, 1]] ]; a[n_] := Mod[d[n], 2]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Apr 22 2015 *)

Prog

(Haskell)
a165560 = flip mod 2 . a003415  -- Reinhard Zumkeller, Mar 11 2014
(Python)
from sympy import factorint
def A165560(n): return int(n&3==2 or (n&1 and sum(factorint(n).values())&1)) # Chai Wah Wu, Nov 04 2022
(PARI) A165560(n) = if(n<=1, 0, my(f=factor(n)); (n*sum(i=1, #f~, f[i, 2]/f[i, 1]))%2); \\ Antti Karttunen, Nov 04 2022

Crossrefs

Characteristic function of A235991, whose complement A235992 gives the positions of 0's.

Cf. A000035, A003415, A347870 [= a(sigma(n))], A353493, A353494, A353495, A358680 (one's complement), A359792.

Sum of A358771 and A358773.

Sequence in context: A126564 A180433 A358771 * A358220 A354874 A014306

Adjacent sequences: A165557 A165558 A165559 * A165561 A165562 A165563

Keyword

easy,nonn

Author

Paolo P. Lava & Giorgio Balzarotti, Sep 24 2009

Extensions

Entries checked by R. J. Mathar, Oct 07 2009

Status

approved