A328893 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.

0, 0, 1, 2, 5, 6, 11, 12, 19, 23, 30, 31, 44, 45, 54, 62, 77, 78, 95, 96, 115, 125, 138, 139, 168, 174, 189, 202, 227, 228, 259, 260, 291, 305, 324, 336, 379, 380, 401, 417, 460, 461, 502, 503, 540, 569, 594, 595, 656, 664, 701, 721, 764, 765, 818, 834, 891, 913

Offset: 0

Author

Daniel Suteu, Oct 29 2019

Keywords

• nonn

Comments

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).

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).

Crossrefs

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

Programs

PARI

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

PARI

ppcount(n) = sum(k=1, logint(n,2), primepi(sqrtnint(n, k))); \\ A025528
f(n) = n*(n+1)/2; \\ A000217
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);

Formula

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

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

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

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