A025528 -id:A025528 - cp's OEIS frontend

A328003 a(n) = ppi(2n) - ppi(n). Number of prime powers (A246655) in the interval (n, 2n]. See comments.

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 5, 5, 6, 7, 8, 7, 7, 7, 7, 8, 9, 8, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 12, 12, 13, 13, 12, 12, 12, 12, 13, 14, 13, 14, 15, 15, 16, 16, 15, 15, 15, 15, 16, 17, 17, 18, 17, 17, 18, 19, 18, 18

Offset: 0

Peter Luschny, Nov 18 2019

nonn

The sequence counts prime powers greater than n up to and including 2*n. - Harvey P. Dale, Aug 01 2020

Cf. A108954, A025528, A246655.

Table[Count[Range[n+1,2n],?PrimePowerQ],{n,0,80}] (* _Harvey P. Dale, Aug 01 2020 *)

def a(n) : return sum([1 for k in (n+1..2*n) if is_prime_power(k)])
print([a(n) for n in (0..72)])

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.

Original entry on oeis.org

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

Daniel Suteu, Oct 29 2019

nonn

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).

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

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

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);

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).

cp's OEIS Frontend

A328003 a(n) = ppi(2n) - ppi(n). Number of prime powers (A246655) in the interval (n, 2n]. See comments.

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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.

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cp's OEIS Frontend

A328003 a(n) = ppi(2*n) - ppi(n). Number of prime powers (A246655) in the interval (n, 2*n]. See comments.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

SageMath

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

Formula

A328003 a(n) = ppi(2n) - ppi(n). Number of prime powers (A246655) in the interval (n, 2n]. See comments.