A064602 -id:A064602 - cp's OEIS frontend

A364970 a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

1, 5, 12, 26, 42, 73, 102, 152, 204, 278, 345, 464, 556, 693, 835, 1021, 1175, 1422, 1613, 1907, 2173, 2496, 2773, 3228, 3569, 4015, 4445, 4998, 5434, 6120, 6617, 7331, 7965, 8717, 9391, 10392, 11096, 12031, 12909, 14059, 14921, 16219, 17166, 18489, 19711, 21072, 22201

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A007437.

Cf. A006218, A024916, A064602, A365409, A365439.

Table[Sum[Binomial[Floor[n/k+2],3],{k,n}],{n,50}] (* Harvey P. Dale, Aug 04 2024 *)

a(n) = sum(k=1, n, binomial(n\k+2, 3));

from math import isqrt
def A364970(n): return (-(s:=isqrt(n))**2*(s+1)*(s+2)+sum((q:=n//k)*(3*k*(k+1)+(q+1)*(q+2)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+1,2) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^3 = 1/(1-x) * Sum_{k>=1} binomial(k+1,2) * x^k/(1-x^k).
a(n) = (A064602(n)+A024916(n))/2. - Chai Wah Wu, Oct 26 2023

A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

Original entry on oeis.org

1, 7, 23, 64, 135, 282, 493, 864, 1375, 2166, 3168, 4715, 6536, 9132, 12278, 16525, 21371, 27998, 35314, 44995, 55847, 69504, 84455, 103882, 124428, 150005, 177921, 212017, 247978, 292890, 339267, 395874, 455796, 526692, 600788, 691066, 782457, 891048, 1004814

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A073570.

Cf. A006218, A024916, A064602, A064603, A064604, A364970, A365409.

a(n) = sum(k=1, n, binomial(n\k+4, 5));

from math import isqrt, comb
def A365439(n): return (-(s:=isqrt(n))**2*comb(s+4,4)+sum((q:=n//k)*(5*comb(k+3,4)+comb(q+4,4)) for k in range(1,s+1)))//5 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+3,4) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1-x^k).
a(n) = (A064604(n)+6*A064603(n)+11*A064602(n)+6*A024916(n))/24. - Chai Wah Wu, Oct 26 2023

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 5, 15, 36, 71, 131, 216, 346, 511, 756, 1042, 1441, 1907, 2527, 3207, 4128, 5097, 6371, 7737, 9442, 11213, 13538, 15848, 18734, 21744, 25423, 29077, 33743, 38238, 43818, 49440, 56104, 62694, 70979, 78749, 88154, 97580, 108790, 119450, 132680, 145021, 159974

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363607.
Cf. A366968, A366969, A366970.
Cf. A024916, A064602, A064603, A366967.

a(n) = sum(k=3, n, binomial(k, 3)*(n\k));

from math import isqrt, comb
def A366971(n): return -comb((s:=isqrt(n))+1,4)*(s+1)+sum(comb((q:=n//w)+1,4)+(q+1)*comb(w,3) for w in range(1,s+1)) # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k)^4 = 1/(1-x) * Sum_{k>=3} binomial(k,3) * x^k/(1-x^k).

a(n) = (A064603(n) - 3*A064602(n) + 2*A024916(n))/6. - Chai Wah Wu, Oct 30 2023

A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).

Original entry on oeis.org

1, 34, 278, 1335, 4461, 12513, 29321, 63146, 122439, 225597, 386649, 644557, 1015851, 1570515, 2333259, 3415660, 4835518, 6792187, 9268287, 12572469, 16673621, 21988337, 28424681, 36677981, 46446732, 58699434, 73107634, 90873690, 111384840, 136555392

Offset: 1

Wesley Ivan Hurt, Sep 30 2014

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001160 (sigma_5).
Cf. A024916: Partial sums of sigma(n)   = A000203(n).
Cf. A064602: Partial sums of sigma_2(n) = A001157(n).
Cf. A064603: Partial sums of sigma_3(n) = A001158(n).
Cf. A064604: Partial sums of sigma_4(n) = A001159(n).

[(&+[DivisorSigma(5,j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Nov 07 2018

with(numtheory): A248076:=n->add(sigma[5](i), i=1..n): seq(A248076(n), n=1..50);

Table[Sum[DivisorSigma[5, i], {i, n}], {n, 30}]
Accumulate[DivisorSigma[5, Range[30]]] (* Vaclav Kotesovec, Mar 30 2018 *)

lista(nn) = vector(nn, n, sum(i=1, n, sigma(i, 5))) \\ Michel Marcus, Sep 30 2014

from math import isqrt
def A248076(n): return ((s:=isqrt(n))**3*(s+1)**2*(1-2*s*(s+1)) + sum((q:=n//k)*(12*k**5+q*(q**2*(q*(2*q+6)+5)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{i=1..n} sigma_5(i) = Sum_{i=1..n} A001160(i).
a(n) ~ Zeta(6) * n^6 / 6. - Vaclav Kotesovec, Sep 02 2018
a(n) ~ Pi^6 * n^6 / 5670. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..n} (Bernoulli(6, floor(1 + n/k)) - 1/42)/6, where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 07 2018
a(n) = Sum_{k=1..n} k^5 * floor(n/k). - Daniel Suteu, Nov 08 2018

A279847 a(n) = Sum_{k=1..n} k^2*(floor(n/k) - 1).

Original entry on oeis.org

0, 1, 2, 7, 8, 22, 23, 44, 54, 84, 85, 151, 152, 206, 241, 326, 327, 458, 459, 605, 664, 790, 791, 1065, 1091, 1265, 1356, 1622, 1623, 2023, 2024, 2365, 2496, 2790, 2865, 3480, 3481, 3847, 4026, 4636, 4637, 5373, 5374, 6000, 6341, 6875, 6876, 7982, 8032, 8787, 9086, 9952, 9953, 11137, 11284

Offset: 1

Ilya Gutkovskiy, Dec 20 2016

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

Total volume of all rectangular prisms with dimensions (x, x, z) and integers x and y, such that x + y = n, 0 < x <= y, and z = floor(y/x). - Wesley Ivan Hurt, Dec 21 2020

			For n = 7 the proper divisors of the first seven positive integers are {0}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1} so a(7) = 0^2 + 1^2 + 1^2 + 1^2 + 2^2 + 1 ^2 + 1^2 + 2^2 + 3^2 + 1^2 = 23.

Index entries for sequences related to sums of divisors

Cf. A000290, A000330, A001157, A064602, A067558, A153485.

Table[Sum[k^2 (Floor[n/k] - 1), {k, 1, n}], {n, 55}]
Table[Sum[DivisorSigma[2, k] - k^2, {k, 1, n}], {n, 55}]

a(n) = sum(k=1, n, k^2*(floor(n/k)-1)) \\ Felix Fröhlich, Dec 20 2016

from math import isqrt
def A279847(n): return (-n*(n+1)*(2*n+1)-(s:=isqrt(n))**2*(s+1)*(2*s+1) + sum((q:=n//k)*(6*k**2+q*(2*q+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

G.f.: -x*(1 + x)/(1 - x)^4 + (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k).
a(n) = A064602(n) - A000330(n).
a(n) = Sum_{k=1..n} A067558(k).
a(n) = Sum_{k=1..n} (A001157(k) - A000290(k)).
a(p^k) = a(p^k-1) + (p^(2*k) - 1)/(p^2 - 1), when p is prime.
a(n) ~ ((zeta(3) - 1)/3)*n^3.
a(n) = Sum_{k=1..floor(n/2)} k^2 * floor((n-k)/k). - Wesley Ivan Hurt, Dec 21 2020

A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 6, 17, 42, 78, 149, 234, 379, 555, 815, 1102, 1557, 2013, 2662, 3388, 4349, 5319, 6695, 8026, 9846, 11712, 14027, 16328, 19503, 22464, 26200, 30030, 34759, 39255, 45221, 50678, 57623, 64465, 72579, 80469, 90665, 99805, 111020, 122146, 135566, 147908, 163638

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A059358.

Cf. A006218, A024916, A064602, A064603, A364970, A365439.

a(n) = sum(k=1, n, binomial(n\k+3, 4));

from math import isqrt, comb
def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).
a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - Chai Wah Wu, Oct 26 2023

A356038 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_2(k).

Original entry on oeis.org

1, 7, 28, 95, 286, 802, 2143, 5519, 13807, 33762, 81060, 191678, 447396, 1032647, 2360593, 5351231, 12041764, 26920297, 59829006, 132262550, 290990077, 637429514, 1390811841, 3023647046, 6551547161, 14151910442, 30481920523, 65480947739, 140318385088, 299995596747

Offset: 1

Vaclav Kotesovec, Jul 24 2022

nonn

Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A001157, A160399, A185003, A356039.

Cf. A006218, A024916, A064602, A064603.

with(numtheory): seq(add(sigma[2](i)*binomial(n,i),i=1..n), n=1..60); # Ridouane Oudra, Oct 25 2022

Table[Sum[Binomial[n, k] * DivisorSigma[2, k], {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 2)); \\ Michel Marcus, Jul 24 2022

a(n) ~ zeta(3) * n^2 * 2^(n-2).

a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^2)*binomial(n,i*j). - Ridouane Oudra, Oct 25 2022

A366915 a(n) = Sum_{k=1..n} (-1)^kk^2floor(n/k).

Original entry on oeis.org

-1, 2, -8, 11, -15, 15, -35, 48, -43, 35, -87, 103, -67, 83, -177, 162, -128, 145, -217, 277, -223, 143, -387, 443, -208, 302, -518, 432, -410, 370, -592, 771, -449, 421, -879, 850, -520, 566, -1134, 1024, -658, 842, -1008, 1310, -1056, 534, -1676, 1714, -737

Offset: 1

Chai Wah Wu, Oct 28 2023

sign

Cf. A024919, A064602.

a[n_]:=Sum[ (-1)^k*k^2*Floor[n/k],{k,n}]; Array[a,49] (* Stefano Spezia, Oct 29 2023 *)

a(n) = sum(k=1, n, (-1)^k*k^2*(n\k)); \\ Michel Marcus, Oct 29 2023

from math import isqrt
def A366915(n): return (-(t:=isqrt(m:=n>>1))**2*(t+1)*((t<<1)+1)+sum((q:=m//k)*(6*k**2+q*((q<<1)+3)+1) for k in range(1,t+1))<<2)//3+((s:=isqrt(n))**2*(s+1)*((s<<1)+1)-sum((q:=n//k)*(6*k**2+q*((q<<1)+3)+1) for k in range(1,s+1)))//6

a(n) = 8*A064602(floor(n/2))-A064602(n).

A366970 a(n) = Sum_{k=3..n} binomial(k-1,2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 36, 60, 89, 131, 176, 245, 311, 404, 502, 631, 751, 926, 1079, 1295, 1501, 1756, 1987, 2330, 2612, 2978, 3332, 3779, 4157, 4707, 5142, 5736, 6278, 6926, 7508, 8336, 8966, 9785, 10555, 11533, 12313, 13427, 14288, 15449, 16521, 17742, 18777, 20306

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363610.

Cf. A006218, A024916, A064602, A366968, A366969, A366971.

a(n) = sum(k=3, n, binomial(k-1, 2)*(n\k));

from math import isqrt
def A366970(n): return (-(s:=isqrt(n))*(s*(s**2-(s<<1)-1)+8)+sum(((q:=n//w)+1)*(q*(q-4)+3*(w**2-3*w+4)) for w in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k)^3 = 1/(1-x) * Sum_{k>=3} binomial(k-1,2) * x^k/(1-x^k).

a(n) = (A064602(n)-3*A024916(n))/2 + A006218(n). - Chai Wah Wu, Oct 30 2023

A330449 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)^2).

Original entry on oeis.org

1, 6, 36, 282, 2460, 25506, 299796, 3921882, 56977740, 913248786, 15917884356, 299358495882, 6066180049020, 131932872768066, 3057940695635316, 75151035318996282, 1954299203147952300, 53684552455571903346, 1553161560008013680676, 47162101103528811791082

Offset: 1

Ilya Gutkovskiy, Dec 15 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A000219, A001157, A008277, A064602, A306046, A318250, A330353, A330450.

nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} k * log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306046.
G.f.: Sum_{k>=1} (k - 1)! * sigma_2(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma_2 = A001157.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000219.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma_2(k).
a(n) ~ n! * zeta(3) * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Dec 15 2019

cp's OEIS Frontend

A364970 a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

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A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

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A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

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A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).

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A279847 a(n) = Sum_{k=1..n} k^2*(floor(n/k) - 1).

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A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

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A356038 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_2(k).

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A366915 a(n) = Sum_{k=1..n} (-1)^k*k^2*floor(n/k).

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A366915 a(n) = Sum_{k=1..n} (-1)^kk^2floor(n/k).