A064602 -id:A064602 - cp's OEIS frontend

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

1, 12, 40, 121, 207, 473, 649, 1142, 1611, 2401, 2853, 4647, 5285, 6879, 8759, 11452, 12558, 16739, 18127, 23353, 27129, 31171, 33219, 43573, 47524, 53210, 59538, 69996, 73274, 89694, 93446, 107195, 116731, 126545, 137505, 164580, 169946, 182244, 195644, 225454

Offset: 1

Seiichi Manyama, Dec 15 2021

nonn

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

Cf. A318742, A350108, A350123, A350125.

Cf. A064602, A143128, A319085.

Accumulate[Table[DivisorSigma[2, k] - 3*k*DivisorSigma[1, k] + 3*k^2*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 17 2021 *)

PARI
```
a(n) = sum(k=1, n, k^2*(n\k)^3);
```

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^3-(d-1)^3)/d^2));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))

from math import isqrt
def A350124(n): return (-(s:=isqrt(n))**4*(s+1)*(2*s+1) + sum((q:=n//k)*(k*(3*(k-1))+q*(k*(9*(k-1))+q*(k*(12*k-6)+2)+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^3 - (d - 1)^3)/d^2.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + x^k)/(1 - x^k)^3.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A064602(n) - 3*A143128(n) + 3*A319085(n).
a(n) ~ n^3 * (log(n) + 2*gamma + (zeta(3) - 1)/3 - Pi^2/6), where gamma is the Euler-Mascheroni constant A001620. (End)

A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

Original entry on oeis.org

1, 6, 34, 295, 3421, 50109, 873653, 17651130, 405071647, 10405074777, 295716745389, 9211817240589, 312086923832843, 11424093750214407, 449317984131076935, 18896062057857406028, 846136323944194170206, 40192544399241119212807

Offset: 1

Seiichi Manyama, Jul 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Partial sums of A062796.

Cf. A006218, A024916, A064602, A064603, A064604, A248076, A319194.

PARI
```
a(n) = sum(k=1, n, n\k*k^k);
```
PARI
```
a(n) = sum(k=1, n, sumdiv(k, d, d^d));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-x^k))/(1-x))

def A355887(n): return n*(1+n**(n-1))+sum(k**k*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

a(n) = Sum_{k=1..n} Sum_{d|k} d^d.

G.f.: (1/(1-x)) * Sum_{k>0} (k * x)^k/(1 - x^k).

A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

Original entry on oeis.org

0, 1, 4, 11, 21, 40, 61, 96, 135, 191, 246, 337, 415, 528, 646, 801, 937, 1145, 1316, 1568, 1802, 2089, 2342, 2737, 3047, 3451, 3841, 4338, 4744, 5358, 5823, 6474, 7060, 7758, 8384, 9294, 9960, 10835, 11657, 12717, 13537, 14739, 15642, 16881, 18025, 19314, 20395

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A069153.
Cf. A002541, A236632.
Cf. A024916, A064602, A366971.

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

from math import isqrt
def A366967(n): return ((s:=isqrt(n))**2*(1-s**2)+sum((q:=n//k)*(3*k*(k-1)+q**2-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

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

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

A109694 Decimal expansion of Sum_{n>=1} 1/sigma_2(n).

Original entry on oeis.org

1, 5, 3, 7, 8, 1, 2, 8, 9, 1, 8, 2, 7, 2, 5, 6, 1, 6, 2, 5, 3, 8, 6, 6, 1, 0, 0, 2, 7, 3, 8, 2, 6, 8, 3, 3, 0, 9, 1, 9, 3, 6, 0, 0, 4, 9, 4, 7, 3, 2, 2, 3, 5, 4, 9, 2, 9, 6, 1, 7, 6, 8, 9, 6, 5, 9, 4, 2, 6, 3, 3, 0, 4, 4, 5, 3, 1, 0, 4, 1, 0, 2, 3, 9, 5, 0, 1, 5, 5, 6, 7, 6, 0, 5, 3, 2, 3, 5, 9, 4, 8, 9, 1, 6, 0

Offset: 1

Franklin T. Adams-Watters, Aug 07 2005

			1.5378128918272561625386610027382683309193600494732235492961768965942633...

Cf. A001157 (sigma_2), A064602.

$MaxExtraPrecision = 1000; Do[Clear[f]; f[p_] := (1 + Sum[(p^2 - 1)/(p^(2*e + 2) - 1), {e, 1, emax}]); m = 1000; cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2]*Exp[N[Sum[cc[[n]] * (PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {emax, 100, 400, 100}] (* Vaclav Kotesovec, Sep 19 2020 *)

N=10^9; prodeuler(p=2,N, sum(k=1,200/log(p),if(k==1,1.,1./((p^(2*k)-1)/(p^2-1))))) \\ The output is 1.537812891756...

Product_p Sum_{k>=0} 1/sigma_2(p^k).

More digits from Vaclav Kotesovec, Sep 19 2020

A175199 a(n) is the smallest integer k such that sigma_2(k) = sigma_2(k + 2n), where sigma_2(k) is the sum of squares of divisors of k (A001157).

Original entry on oeis.org

24, 430, 645, 860, 120, 864, 168, 1720, 1935, 10790, 264, 2580, 2795, 1570, 16185, 3440, 408, 3870, 456, 21580, 2355, 4730, 552, 5160, 600, 5590, 5805, 3140, 696, 4320, 744, 6880, 7095, 1248, 840, 7740, 888, 8170, 8385, 43160, 984, 4710, 1032, 9460

Offset: 1

Michel Lagneau, Mar 03 2010

nonn

The equation sigma_2(n) = sigma_2(n + p) has infinitely many solutions where p >= 2 and p is even (J. M. De Koninck).

			For n=1, sigma_2(24) = sigma_2(26) = 850.
For n=2, sigma_2(430) = sigma_2(434) = 240500.
For n=3, sigma_2(645) = sigma_2(651) = 481000.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Amiram Eldar, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. M. De Koninck, On the solutions of sigma_2(n) = sigma_2(n + p), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001158, A001159.

Cf. A053807, A064602.

with(numtheory):for k from 2 by 2 to 200 do :indic:=0:for n from 1 to 100000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+k):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 and indic=0 then print(k):print(n):indic:=1:else fi:od:od:

Edited by Robert Israel, Aug 02 2024

A309176 a(n) = n^2 * (n + 1)/2 - Sum_{k=1..n} sigma_2(k).

Original entry on oeis.org

0, 0, 2, 3, 12, 13, 33, 40, 66, 81, 135, 135, 212, 249, 319, 354, 489, 511, 681, 725, 876, 981, 1233, 1235, 1509, 1660, 1920, 2032, 2437, 2472, 2936, 3091, 3488, 3755, 4275, 4290, 4955, 5292, 5854, 6024, 6843, 6968, 7870, 8190, 8839, 9340, 10420, 10442, 11568, 12038, 13014, 13474, 14851, 15098, 16436

Offset: 1

Ilya Gutkovskiy, Jul 15 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000326, A001157, A004125, A048158, A051126, A154585, A256532.

Cf. A002411, A064602.

Table[n^2 (n + 1)/2 - Sum[DivisorSigma[2, k], {k, 1, n}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[x (1 + 2 x)/(1 - x)^4 - 1/(1 - x) Sum[k^2 x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[Mod[n, k] k, {k, 1, n}], {n, 1, 55}]

a(n) = n^2*(n+1)/2 - sum(k=1, n, sigma(k, 2)); \\ Michel Marcus, Sep 18 2021

from math import isqrt
def A309176(n): return (n**2*(n+1)>>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 + 2*x)/(1 - x)^4 - (1/(1 - x)) * Sum_{k>=1} k^2 * x^k/(1 - x^k).
a(n) = Sum_{k=1..n} (n mod k) * k.
a(n) = A002411(n) - A064602(n).

A379921 Partial alternating sums of the sigma_2 function: a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_2(k).

Original entry on oeis.org

1, -4, 6, -15, 11, -39, 11, -74, 17, -113, 9, -201, -31, -281, -21, -362, -72, -527, -165, -711, -211, -821, -291, -1141, -490, -1340, -520, -1570, -728, -2028, -1066, -2431, -1211, -2661, -1361, -3272, -1902, -3712, -2012, -4222, -2540, -5040, -3190, -5752, -3386

Offset: 1

Amiram Eldar, Jan 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157 (sigma_2), A002117, A064602, A068762, A307704.

Accumulate[Table[(-1)^(k+1) * DivisorSigma[2, k], {k, 1, 100}]]

list(lim) = {my(s = 0); for(k = 1, lim, s += (-1)^(k+1) * sigma(k, 2); print1(s, ", "));}

a(n) ~ -zeta(3) * n^3 / 24.

In general, for m >= 2, Sum_{k=1..n} (-1)^(k+1) * sigma_m(k) ~ -zeta(m+1) * n^(m+1) / ((m+1)*2^(m+1)).

A356039 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_3(k).

Original entry on oeis.org

1, 11, 58, 243, 866, 2804, 8485, 24387, 67333, 180086, 469338, 1196976, 2996956, 7385837, 17954243, 43125267, 102494548, 241309031, 563341508, 1305142418, 3002938045, 6866090880, 15609292379, 35299794600, 79443050541, 177989130174, 397124963671, 882642816697, 1954708794400

Offset: 1

Vaclav Kotesovec, Jul 24 2022

nonn

For m>0, Sum_{k=1..n} binomial(n,k) * sigma_m(k) ~ zeta(m+1) * n^m * 2^(n-m).

Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A001158, A160399, A185003, A356038.

Cf. A006218, A024916, A064602, A064603.

with(numtheory): seq(add(sigma[3](i)*binomial(n,i), i=1..n), n=1..60); # Ridouane Oudra, Oct 31 2022

Table[Sum[Binomial[n, k] * DivisorSigma[3, k], {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 3)); \\ Michel Marcus, Jul 24 2022

a(n) ~ Pi^4 * n^3 * 2^(n-4) / 45.

a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^3)*binomial(n,i*j). - Ridouane Oudra, Oct 31 2022

A356339 a(n) = Sum_{k=1..n} binomial(2n, n-k) sigma_2(k).

Original entry on oeis.org

1, 9, 55, 297, 1496, 7215, 33783, 154825, 698077, 3107424, 13690161, 59802471, 259377080, 1118176887, 4795381640, 20472223529, 87051685546, 368857919085, 1558036408998, 6562564601592, 27571934249754, 115574440020477, 483444570596465, 2018365519396135, 8411811012694246

Offset: 1

Vaclav Kotesovec, Aug 04 2022

nonn

Cf. A001157, A064602, A351146, A356038.

Table[Sum[Binomial[2*n, n-k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]

a(n) = sum(k=1, n, binomial(2*n, n-k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022

a(n) ~ zeta(3) * n * 4^(n-1).

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

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

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A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

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A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

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A109694 Decimal expansion of Sum_{n>=1} 1/sigma_2(n).

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A175199 a(n) is the smallest integer k such that sigma_2(k) = sigma_2(k + 2n), where sigma_2(k) is the sum of squares of divisors of k (A001157).

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A309176 a(n) = n^2 * (n + 1)/2 - Sum_{k=1..n} sigma_2(k).

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A379921 Partial alternating sums of the sigma_2 function: a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_2(k).

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A356039 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_3(k).

A356339 a(n) = Sum_{k=1..n} binomial(2n, n-k) sigma_2(k).