A034387 -id:A034387 - cp's OEIS frontend

A301707 a(n) = n * Sum_{k prime<=n} k.

0, 4, 15, 20, 50, 60, 119, 136, 153, 170, 308, 336, 533, 574, 615, 656, 986, 1044, 1463, 1540, 1617, 1694, 2300, 2400, 2500, 2600, 2700, 2800, 3741, 3870, 4960, 5120, 5280, 5440, 5600, 5760, 7289, 7486, 7683, 7880, 9758, 9996, 12083, 12364, 12645, 12926, 15416, 15744

Offset: 1

Pedro Caceres, Mar 25 2018

nonn

			a(6) = 6 * Sum_{k primes<=6} = 6 * (2+3+5) = 6 * 10 = 60.

Cf. A034387.

Array[# Sum[Prime@ i, {i, PrimePi@ #}] &, 48] (* Michael De Vlieger, Apr 10 2018 *)

a(n) = n*sum(k=1, n, k*isprime(k)); \\ Michel Marcus, Mar 26 2018

a(n) = n * A034387(n).

More terms from Michel Marcus, Apr 09 2018

A333700 a(n) = Sum_{k=1..n} pi(k) * pi(n-k).

Original entry on oeis.org

0, 0, 0, 1, 4, 8, 14, 22, 32, 45, 58, 73, 90, 110, 132, 158, 184, 214, 246, 282, 320, 363, 406, 455, 506, 562, 618, 678, 738, 804, 872, 944, 1018, 1099, 1180, 1269, 1358, 1450, 1544, 1644, 1744, 1852, 1962, 2078, 2196, 2321, 2446, 2581, 2718, 2863

Offset: 1

Ilya Gutkovskiy, Apr 02 2020

nonn

Convolution of A000720 with itself.

Mathematics Stack Exchange, A curious equality of integrals involving the prime counting function?
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A000720, A002815, A010051, A034387, A046992, A073610.

Table[Sum[PrimePi[k] PrimePi[n - k], {k, n}], {n, 50}]
nmax = 50; CoefficientList[Series[(1/(1 - x)^2) Sum[x^Prime[k], {k, 1, nmax}]^2, {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, primepi(k)*primepi(n-k)); \\ Michel Marcus, Apr 03 2020

G.f.: (1/(1 - x)^2) * (Sum_{k>=1} x^prime(k))^2.
a(n) = Sum_{k=1..n} A046992(k) * A010051(n-k).
a(n) = Sum_{k=1..n} k * A073610(n-k+1).
From Jianing Song, Sep 27 2023: (Start)
a(n-1) = Integral_{0..n} pi(x) * pi(n-x) dx, since Integral_{0..n} pi(x) * pi(n-x) dx = Sum_{k=1..n} Integral_{k-1..k} pi(x) * pi(n-x) dx = Sum_{k=1..n} pi(k-1) * pi(n-k) = Sum_{k=0..n-1} pi(k) * pi(n-1-k) = a(n-1).
a(n) = (a(n-1) + a(n+1))/2 for n == 4 (mod 6) with n > 4, as shown in the Mathematics Stack Exchange link. (End)

A351827 Sum of the numbers <= n that are either prime, a divisor of n, or both.

Original entry on oeis.org

1, 3, 6, 10, 11, 17, 18, 30, 27, 28, 29, 51, 42, 56, 57, 70, 59, 92, 78, 112, 99, 100, 101, 155, 126, 127, 137, 147, 130, 191, 161, 221, 194, 195, 196, 246, 198, 236, 237, 280, 239, 322, 282, 352, 351, 328, 329, 447, 378, 414, 380, 411, 382, 496, 437, 492, 439, 440, 441, 598

Offset: 1

Wesley Ivan Hurt, Feb 20 2022

nonn

Cf. A000203 (sigma), A008472 (sopf), A010051, A034387, A351519.

a(n) = Sum_{k=1..n} k * (u(k) + v(n/k) - u(k)*v(n/k)), where u(n) is the prime characteristic (A010051) and v(n) = 1 - ceiling(n) + floor(n).

a(n) = sigma(n) - sopf(n) + Sum_{p<=n, p prime} p. - Wesley Ivan Hurt, Dec 31 2023

A351914 Numbers m such that the average of the prime numbers up to m is greater than or equal to m/2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 13, 19

Offset: 1

Masahiko Shin, Feb 25 2022

The average of the prime numbers up to m is asymptotically equal to m/2 by the prime number theorem. It is shown by Mandl's inequality that m/2 is strictly greater than the average if m > 19 and thus the sequence is complete.

			5 is a term since the average of the primes up to 5 is (2 + 3 + 5)/3 = 10/3, which is greater than 5/2.
8 is a term since the average of the primes up to 8 is (2 + 3 + 5 + 7)/4 = 17/4 = 4.25, which is greater than 8/2 = 4.

Hassani Mehdi, A Refinement of Mandl's Inequality, report collection, 8 (2) (2005).
Eric Bach and Jefrey Shallit, Algorithmic Number Theory, Vol. 1: Efficient Algorithms, Section 2.7, Cambridge, MA: MIT Press, 1996.
Math Stackexchange, Is the average of primes up to n smaller than n/2, if n>19?
Matt Visser, On the arithmetic average of the first n primes, arXiv:2505.04951 [math.NT], 2025. See p. 3.

Cf. A000040, A000720, A034387.

s:= proc(n) s(n):= `if`(n=0, 0, `if`(isprime(n), n, 0)+s(n-1)) end:
q:= n-> is(2*s(n)/numtheory[pi](n)>=n):
select(q, [$2..20])[];  # Alois P. Heinz, Feb 25 2022

s[n_] := s[n] = If[n == 0, 0, If[PrimeQ[n], n, 0] + s[n-1]];
Select[Range[2, 20], 2 s[#]/PrimePi[#] > #&] (* Jean-François Alcover, Dec 26 2022, after Alois P. Heinz *)

from sympy import primerange
def average_of_primes_up_to(i):
    primes_up_to_i =  list(primerange(2, i+1))
    return sum(primes_up_to_i) / len(primes_up_to_i)
def a_list():
    return [i for i in range(2, 20) if average_of_primes_up_to(i) >= i / 2]

2*A034387(a(n))/A000720(a(n)) >= a(n).

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A301707 a(n) = n * Sum_{k prime<=n} k.

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A333700 a(n) = Sum_{k=1..n} pi(k) * pi(n-k).

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A351827 Sum of the numbers <= n that are either prime, a divisor of n, or both.

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A351914 Numbers m such that the average of the prime numbers up to m is greater than or equal to m/2.

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