cp's OEIS Frontend

A307743 a(n) = Sum_{k=1..n} A307742(k).

%I A307743 #49 May 13 2019 15:32:37
%S A307743 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 A307743 46,48,48,55,55,62,63,63,63,63,63,70,70,70,70,77,77,85,85,85,85,94,94,
%U A307743 98,98,98,98,106,106,106,106,106,106,115,115,123,123,123,124
%N A307743 a(n) = Sum_{k=1..n} A307742(k).
%C A307743 Quasi-logarithmic analog of the summatory von Mangoldt function, i.e., of the second Chebyshev function.
%C A307743 Conjecture: There is a constant c such that abs(a(n) - 2*n*(c+1)/c) = O(sqrt(n)).
%H A307743 I. V. Serov, <a href="/A307743/b307743.txt">Table of n, a(n) for n = 1..10000</a>
%t A307743 qLog[n_] := qLog[n] = Module[{p, e}, If[n == 1, 0, Sum[{p, e} = pe; (1 + qLog[p - 1]) e, {pe, FactorInteger[n]}]]];
%t A307743 f[n_] := qLog[Exp[MangoldtLambda[n]]];
%t A307743 a[n_] := Sum[f[k], {k, 1, n}];
%t A307743 Array[a, 64] (* _Jean-François Alcover_, May 07 2019 *)
%o A307743 (PARI) mang(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
%o A307743 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 A307743 f(n) = ql(mang(n)); \\ A307742
%o A307743 a(n) = sum(k=1, n, f(k)); \\ _Michel Marcus_, Apr 27 2019
%Y A307743 Cf. A307742, A064097, A014963, A008683.
%K A307743 nonn
%O A307743 1,3
%A A307743 _I. V. Serov_, Apr 26 2019