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

%I A307742 #31 May 15 2019 04:52:26
%S A307742 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,
%T A307742 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,
%U A307742 0,0,9,0,8,0,0,0,0,0,9,0,2,0,9,0,0,0,0,0,9
%N A307742 Quasi-logarithm A064097(n) of von Mangoldt's exponential function A014963(n).
%H A307742 I. V. Serov, <a href="/A307742/b307742.txt">Table of n, a(n) for n = 1..10000</a>
%F A307742 a(n) = A064097(A014963(n)).
%F A307742 a(n) = 1 + A064097(n-1) if n is prime.
%F A307742 a(n) = a(p) if n=p^k with k > 1.
%F A307742 a(n) = 0 if n is not a prime power or n = 1.
%F A307742 a(n) = -Sum_{d|n} A064097(d)*A008683(d) by Mobius inversion.
%t A307742 qLog[n_] := qLog[n] = Module[{p, e}, If[n == 1, 0, Sum[{p, e} = pe; (1 + qLog[p-1])e, {pe, FactorInteger[n]}]]];
%t A307742 a[n_] := qLog[Exp[MangoldtLambda[n]]];
%t A307742 Array[a, 100] (* _Jean-François Alcover_, May 07 2019 *)
%o A307742 (PARI) mang(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
%o A307742 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
%o A307742 a(n) = ql(mang(n)); \\ _Michel Marcus_, Apr 26 2019
%Y A307742 Cf. A064097, A014963, A008683, A307743.
%K A307742 nonn
%O A307742 1,3
%A A307742 _I. V. Serov_, Apr 26 2019