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

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, 46, 48, 48, 55, 55, 62, 63, 63, 63, 63, 63, 70, 70, 70, 70, 77, 77, 85, 85, 85, 85, 94, 94, 98, 98, 98, 98, 106, 106, 106, 106, 106, 106, 115, 115, 123, 123, 123, 124

Offset: 1

Author

I. V. Serov, Apr 26 2019

Keywords

• nonn

Comments

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

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

Links

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

Crossrefs

Cf. A307742, A064097, A014963, A008683.

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]}]]];
f[n_] := qLog[Exp[MangoldtLambda[n]]];
a[n_] := Sum[f[k], {k, 1, n}];
Array[a, 64] (* 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
f(n) = ql(mang(n)); \\ A307742
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, Apr 27 2019