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

1,19

Links

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

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)).

Mathematica

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

Prog

(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])))); };

Crossrefs

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

Cf. A003401 (positions of zeros).

Sequence in context: A140081 A280596 A112345 * A265262 A124763 A029372

Adjacent sequences: A336466 A336467 A336468 * A336470 A336471 A336472

Keyword

nonn

Author

Antti Karttunen, Jul 22 2020

Status

approved