A355947 a(n) = Sum_{k=1..n} (n+1-k)*floor(n/k).

0, 1, 5, 12, 25, 39, 65, 87, 124, 161, 210, 249, 328, 377, 450, 531, 630, 698, 825, 903, 1047, 1169, 1295, 1393, 1609, 1740, 1893, 2056, 2269, 2400, 2679, 2822, 3070, 3277, 3486, 3709, 4082, 4260, 4498, 4748, 5136, 5336, 5744, 5956, 6312, 6686, 6984, 7218, 7772

Offset: 0

Author

Tamas Sandor Nagy, Jul 21 2022

Keywords

• nonn
• easy

Comments

In the sum, the divisors are k = 1..n and the multipliers are the reverse n+1-k = n .. 1.

From Thomas Scheuerle, Jul 21 2022: (Start)

Without the floor operation a(n) would be A027457(n+1)/A099946(n+1) = (H(n+1) - 1)*(n^2+n), where H(n) is the n-th harmonic number.

What is lim_{n->oo} ((A027457(n+1)/A099946(n+1) - a(n))/(n^2+n))? It appears to be close to 0.2452... . (End)

This limit is equal to Pi^2/12 - gamma = 0.24525136852258... - Vaclav Kotesovec, Jul 23 2022

Examples

			For n=5, the sum is formed:
  k = 1..n:                1   2   3   4   5
  floor(n/k):              5   2   1   1   1
  n+1-k = n..1:            5   4   3   2   1
  floor(n/k)*(n+1-k):     25   8   3   2   1
                          __________________
  a(5) =                  25 + 8 + 3 + 2 + 1 = 39
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A006218, A024916, A027457, A099946, A143274.

Programs

Mathematica

a[n_] := Sum[(n+1-k) * Floor[n/k], {k, 1, n}]; Array[a, 50, 0] (* Amiram Eldar, Jul 22 2022 *)

PARI

a(n) = sum(k=1, n, (n+1-k)*floor(n/k)) \\ Rémy Sigrist, Jul 21 2022

PARI

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*(k-(k-1)*x-x^k)/(1-x^k)^2)/(1-x)^2)) \\ Seiichi Manyama, Jul 24 2022

Python

from math import isqrt
def A355947(n): return (s:=isqrt(n))**2*(s-(n<<1)-1)+sum((q:=n//k)*((n<<2)-(k<<1)-q+3) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 24 2023

Formula

From Vaclav Kotesovec, Jul 23 2022: (Start)

For n > 0, a(n) = (n+1)*A006218(n) - A024916(n).

a(n) ~ n*(n+1)*(log(n) + 2*gamma - 1) - n^2*Pi^2/12, where gamma is the Euler-Mascheroni constant A001620. (End)

(1/(1-x)^2) * Sum_{k>0} x^k * (k - (k-1)*x - x^k)/(1-x^k)^2. - Seiichi Manyama, Jul 24 2022

Extensions

More terms from Rémy Sigrist, Jul 21 2022