A328893 - OEIS

%I #5 Nov 06 2019 18:34:51

%S 0,0,1,2,5,6,11,12,19,23,30,31,44,45,54,62,77,78,95,96,115,125,138,

%T 139,168,174,189,202,227,228,259,260,291,305,324,336,379,380,401,417,

%U 460,461,502,503,540,569,594,595,656,664,701,721,764,765,818,834,891,913

%N Partial sums of A095112: sum of the Dirichlet convolution of the characteristic function of the prime powers (A069513) with the positive integers (A000027) from 1 to n.

%C In general, for m >= 0, if we define f(n,m) = Sum_{p^k|n} Sum_{j=1..k} n^m/p^(m*j) (cf. A322664), then Sum_{k=1..n} f(k,m) = Sum_{k=1..n} Sum_{d|k} A069513(k/d) * d^m = Sum_{k=1..n} A069513(k) * F_m(floor(n/k)), where F_m(x) are the Faulhaber polynomials defined as F_m(x) = (Bernoulli(m+1, x+1) - Bernoulli(m+1, 1)) / (m+1).

%C Additionally, for m >= 1, Sum_{k=1..n} f(k,m) ~ Q(m+1) * n^(m+1)/(m+1), where Q(s) = Sum_{p prime} 1/(p^s - 1).

%F a(n) ~ A154945 * n*(n+1)/2.

%F a(n) = Sum_{k=1..n} k * A025528(floor(n/k)).

%F a(n) = Sum_{k=1..n} Sum_{d|k} d * A069513(k/d).

%F a(n) = (1/2)*Sum_{k=1..n} A069513(k) * floor(n/k) * floor(1+n/k).

%o (PARI) a(n) = sum(k=1, n, if(isprimepower(k), (n\k) * (1+n\k), 0))/2;

%o (PARI) ppcount(n) = sum(k=1, logint(n,2), primepi(sqrtnint(n, k))); \\ A025528

%o f(n) = n*(n+1)/2; \\ A000217

%o a(n) = my(s=sqrtint(n)); sum(k=1, s, if(isprimepower(k), f(n\k), 0) + k*ppcount(n\k)) - f(s)*ppcount(s);

%Y Cf. A025528, A022559, A069513, A154945, A322068.

%K nonn

%O 0,4

%A _Daniel Suteu_, Oct 29 2019