A307743 - OEIS

%I #49 May 13 2019 15:32:37

%S 0,1,3,4,7,7,11,12,14,14,19,19,24,24,24,25,30,30,36,36,36,36,43,43,46,

%T 46,48,48,55,55,62,63,63,63,63,63,70,70,70,70,77,77,85,85,85,85,94,94,

%U 98,98,98,98,106,106,106,106,106,106,115,115,123,123,123,124

%N a(n) = Sum_{k=1..n} A307742(k).

%C Quasi-logarithmic analog of the summatory von Mangoldt function, i.e., of the second Chebyshev function.

%C Conjecture: There is a constant c such that abs(a(n) - 2*n*(c+1)/c) = O(sqrt(n)).

%H I. V. Serov, <a href="/A307743/b307743.txt">Table of n, a(n) for n = 1..10000</a>

%t qLog[n_] := qLog[n] = Module[{p, e}, If[n == 1, 0, Sum[{p, e} = pe; (1 + qLog[p - 1]) e, {pe, FactorInteger[n]}]]];

%t f[n_] := qLog[Exp[MangoldtLambda[n]]];

%t a[n_] := Sum[f[k], {k, 1, n}];

%t Array[a, 64] (* _Jean-François Alcover_, May 07 2019 *)

%o (PARI) mang(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963

%o 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 f(n) = ql(mang(n)); \\ A307742

%o a(n) = sum(k=1, n, f(k)); \\ _Michel Marcus_, Apr 27 2019

%Y Cf. A307742, A064097, A014963, A008683.

%K nonn

%O 1,3

%A _I. V. Serov_, Apr 26 2019