A307742 Quasi-logarithm A064097(n) of von Mangoldt's exponential function A014963(n).

0, 1, 2, 1, 3, 0, 4, 1, 2, 0, 5, 0, 5, 0, 0, 1, 5, 0, 6, 0, 0, 0, 7, 0, 3, 0, 2, 0, 7, 0, 7, 1, 0, 0, 0, 0, 7, 0, 0, 0, 7, 0, 8, 0, 0, 0, 9, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 8, 0, 0, 1, 0, 0, 9, 0, 0, 0, 9, 0, 8, 0, 0, 0, 0, 0, 9, 0, 2, 0, 9, 0, 0, 0, 0, 0, 9

Offset: 1

Author

I. V. Serov, Apr 26 2019

Keywords

• nonn

Links

• I. V. Serov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A064097, A014963, A008683, A307743.

Programs

Mathematica

qLog[n_] := qLog[n] = Module[{p, e}, If[n == 1, 0, Sum[{p, e} = pe; (1 + qLog[p-1])e, {pe, FactorInteger[n]}]]];
a[n_] := qLog[Exp[MangoldtLambda[n]]];
Array[a, 100] (* Jean-François Alcover, May 07 2019 *)

PARI

mang(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
ql(n) = if (n==1, 0, if(isprime(n),1+ql(n-1), sumdiv(n,p, if(isprime(p),ql(p)*valuation(n,p))))); \\ A064097
a(n) = ql(mang(n)); \\ Michel Marcus, Apr 26 2019

Formula

a(n) = A064097(A014963(n)).

a(n) = 1 + A064097(n-1) if n is prime.

a(n) = a(p) if n=p^k with k > 1.

a(n) = 0 if n is not a prime power or n = 1.

a(n) = -Sum_{d|n} A064097(d)*A008683(d) by Mobius inversion.