A353493 The arithmetic derivative of n, reduced modulo 4.

0, 0, 1, 1, 0, 1, 1, 1, 0, 2, 3, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 3, 3, 0, 1, 3, 1, 0, 2, 3, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 3, 1, 1, 0, 2, 1, 0, 0, 1, 1, 0, 0, 2, 3, 1, 0, 1, 1, 3, 0, 2, 1, 1, 0, 2, 3, 1, 0, 1, 3, 3, 0, 2, 3, 1, 0, 0, 3, 1, 0, 2, 1, 0, 0, 1, 3, 0, 0, 2, 1, 0, 0, 1, 1, 3, 0, 1, 3, 1, 0, 3

Offset: 0

Author

Antti Karttunen, Apr 22 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A003415, A010873, A165560, A235991, A353490, A353494, A353495, A353496, A353497.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A353493(n) = (A003415(n)%4);

Formula

a(n) = A010873(A003415(n)).

For all n, a(4*n) = 0 and a(4*n + 2) is either 1 or 3. [See comments in A235991]

For all n >= 2, a(n) = A010873[(A353496(n)*A353497(n)) + A353490(n)]. (This is essentially Reinhard Zumkeller's May 09 2011 recursive formula of A003415, when reduced modulo 4) - Antti Karttunen, Apr 26 2022