A279847 -id:A279847 - cp's OEIS frontend

A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2k) floor((n-k)/k).

0, 0, 2, 6, 13, 24, 37, 56, 78, 106, 132, 178, 212, 258, 312, 376, 425, 508, 565, 662, 749, 836, 909, 1058, 1156, 1264, 1384, 1536, 1636, 1836, 1946, 2126, 2282, 2434, 2606, 2880, 3019, 3194, 3385, 3676, 3833, 4138, 4305, 4572, 4863, 5086, 5271, 5692, 5924, 6240

Offset: 1

Wesley Ivan Hurt, Dec 21 2020

Total area of all rectangles with dimensions (y-x) X (z) where x and y are integers such that x + y = n, 0 < x <= y, and z = floor(y/x).

Cf. A001620, A002541, A006218, A024916, A153485, A279847, A339370.

Table[Sum[(n - 2 k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 60}]

from math import isqrt
def A338991(n): return ((s:=isqrt(n))+1)*(n*(1-s)+s**2)-sum((q:=n//k)*((k-n<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 23 2023

From Vaclav Kotesovec, Jun 24 2021: (Start)
a(n) = n + n*A006218(n) - 2*A024916(n).
a(n) ~ (log(n) + 2*gamma - Pi^2/6 - 1)*n^2, where gamma is the Euler-Mascheroni constant A001620. (End)

A339217 a(n) = Sum_{k=1..n} floor((2*n-k)/k).

Original entry on oeis.org

0, 1, 4, 8, 12, 17, 23, 27, 34, 40, 46, 52, 60, 65, 73, 81, 87, 93, 104, 108, 118, 126, 132, 140, 150, 157, 165, 173, 183, 189, 201, 205, 216, 226, 232, 242, 254, 258, 268, 278, 288, 295, 307, 313, 323, 335, 343, 349, 363, 369, 382, 390, 398, 408, 420, 428, 440, 448, 456, 464, 482

Offset: 0

Wesley Ivan Hurt, Dec 22 2020

Cf. A002541, A153485, A279847, A338991, A339370.

Table[Sum[Floor[(2 n - i)/i], {i, n}], {n, 0, 60}]

a(n) = sum(k=1, n, (2*n-k)\k); \\ Michel Marcus, Dec 22 2020

From Vaclav Kotesovec, Dec 23 2020: (Start)
For n>0, a(n) = 2*A006218(n) + A075989(n) - n.
a(n) ~ 2*n * (log(2*n) + 2*gamma - 2), where gamma is the Euler-Mascheroni constant A001620. (End)

A339804 a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * floor((n-k)/k).

Original entry on oeis.org

0, 1, 4, 13, 22, 50, 68, 116, 162, 236, 278, 437, 498, 634, 794, 1018, 1118, 1450, 1574, 1975, 2276, 2598, 2774, 3519, 3834, 4273, 4746, 5490, 5772, 6887, 7214, 8163, 8856, 9586, 10330, 12072, 12540, 13443, 14382, 16244, 16806, 18861, 19480, 21192, 22954, 24267

Offset: 1

Wesley Ivan Hurt, Dec 17 2020

Total volume of all rectangular prisms with dimensions (x, y, z) where x and y are positive integers such that x + y = n, x <= y, and z = floor(y/x). - Wesley Ivan Hurt, Dec 20 2020

Cf. A002541, A153485, A279847, A339370.

Table[Sum[k (n - k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 50}]

a(n) = sum(k=1, n\2, k*(n-k)*((n-k)\k)); \\ Michel Marcus, Dec 19 2020

from math import isqrt
def A339804(n): return (n*(1-n**2)+((s:=isqrt(n))**4<<1)+s**3*(3*(1-n))+s**2*(1-3*n) + sum((q:=n//k)*(-6*k**2+n*(3*((k<<1)+q+1))-q*((q<<1)+3)-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 27 2023

a(n) ~ n^3*(Pi^2-2-4*zeta(3))/12. - Rok Cestnik, Dec 19 2020

a(n) = n*A153485(n) - A279847(n). - Vaclav Kotesovec, Dec 21 2020

A280385 a(n) = Sum_{k=1..n} prime(k)^2*floor(n/prime(k)) .

Original entry on oeis.org

0, 4, 13, 17, 42, 55, 104, 108, 117, 146, 267, 280, 449, 502, 536, 540, 829, 842, 1203, 1232, 1290, 1415, 1944, 1957, 1982, 2155, 2164, 2217, 3058, 3096, 4057, 4061, 4191, 4484, 4558, 4571, 5940, 6305, 6483, 6512, 8193, 8255, 10104, 10229, 10263, 10796, 13005, 13018, 13067, 13096, 13394, 13567, 16376, 16389, 16535

Offset: 1

Ilya Gutkovskiy, Jan 01 2017

Sum of all squares of prime divisors of all positive integers <= n.

Partial sums of A005063.

			For n = 6 the prime divisors of the first six positive integers are {0}, {2}, {3}, {2}, {5}, {2, 3} so a(6) = 0^2 + 2^2 + 3^2 + 2^2 + 5^2 + 2^2 + 3^2 = 55.

Index entries for sequences related to sums of divisors

Cf. A001157, A005063, A008472, A024924, A064602, A279847, A279910.

Table[Sum[Prime[k]^2 Floor[n/Prime[k]], {k, 1, n}], {n, 55}]
Table[Sum[DivisorSum[k, #1^2 &, PrimeQ[#1] &], {k, 1, n}], {n, 55}]
nmax = 55; Rest[CoefficientList[Series[(1/(1 - x)) Sum[Prime[k]^2 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sum(k=1, n, prime(k)^2 * (n\prime(k))); \\ Indranil Ghosh, Apr 03 2017

from sympy import prime
print([sum([prime(k)**2 * (n//prime(k)) for k in range(1, n + 1)]) for n in range(1, 21)]) # Indranil Ghosh, Apr 03 2017

G.f.: (1/(1 - x))*Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)).

cp's OEIS Frontend

A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2*k) * floor((n-k)/k).

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A339217 a(n) = Sum_{k=1..n} floor((2*n-k)/k).

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A339804 a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * floor((n-k)/k).

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A280385 a(n) = Sum_{k=1..n} prime(k)^2*floor(n/prime(k)) .

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Formula

A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2k) floor((n-k)/k).