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

Author

Paolo P. Lava & Giorgio Balzarotti, Sep 24 2009

Keywords

• easy
• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for characteristic functions

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.

Programs

Haskell

a165560 = flip mod 2 . a003415  -- Reinhard Zumkeller, Mar 11 2014

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 *)

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

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

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

Extensions

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