A336469 a(n) = A329697(phi(n)), where A329697 is totally additive with a(2) = 0 and a(p) = 1 + a(p-1) for odd primes.

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

Offset: 1

Author

Antti Karttunen, Jul 22 2020

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A000265, A003434, A005087, A046660, A064097, A064415, A329697, A336396, A336477.

Cf. A003401 (positions of zeros).

Programs

Mathematica

Array[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, EulerPhi[#], # != 2^IntegerExponent[#, 2] &] - 1 &, 105] (* Michael De Vlieger, Jul 24 2020 *)

PARI

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A336469(n) = A329697(eulerphi(n));
\\ Or alternatively as:
A336469(n) = { my(f = factor(n)); sum(k=1, #f~, if(2==f[k,1],0,-1 + (f[k, 2]*A329697(f[k, 1])))); };

Formula

Additive with a(2^e) = 0, and for odd primes p, a(p^e) = A329697((p - 1)*p^(e-1)) = e*A329697(p) - 1.

a(n) = A329697(n) - A005087(n) = A336396(n) + A046660(A000265(n)).