A248076 -id:A248076 - cp's OEIS frontend

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

1, 6, 16, 37, 63, 113, 163, 248, 339, 469, 591, 801, 971, 1221, 1481, 1822, 2112, 2567, 2929, 3475, 3975, 4585, 5115, 5965, 6616, 7466, 8286, 9336, 10178, 11478, 12440, 13805, 15025, 16475, 17775, 19686, 21056, 22866, 24566, 26776, 28458, 30958

Offset: 1

Labos Elemer, Sep 24 2001

nonn

In general, for m >= 0 and j >= 0, Sum_{k=1..n} k^m * sigma_j(k) = Sum_{k=1..s} (k^m * F_{m+j}(floor(n/k)) + k^(m+j) * F_m(floor(n/k))) - F_{m+j}(s) * F_m(s), where s = floor(sqrt(n)) and F_m(x) are the Faulhaber polynomials defined as F_m(x) = (Bernoulli(m+1, x+1) - Bernoulli(m+1, 1)) / (m+1). - Daniel Suteu, Nov 27 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 401: Sum of squares of divisors, 2012.

Cf. A001157, A064605.

Cf. A064603, A064604, A248076.

Accumulate@ Array[DivisorSigma[2, #] &, 42] (* Michael De Vlieger, Jan 02 2017 *)

a(n) = sum(j=1, n, sigma(j, 2)); \\ Michel Marcus, Dec 15 2013

f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
a(n) = my(s=sqrtint(n)); sum(k=1, s, f(n\k) + k^2*(n\k)) - s*f(s); \\ Daniel Suteu, Nov 26 2020

from math import isqrt
def f(n): return n*(n+1)*(2*n+1)//6
def a(n):
    s = isqrt(n)
    return sum(f(n//k) + k*k*(n//k) for k in range(1, s+1)) - s*f(s)
print([a(k) for k in range(1, 43)]) # Michael S. Branicky, Oct 01 2022 after Daniel Suteu

a(n) = a(n-1) + A001157(n) = Sum_{j=1..n} sigma_2(j) where sigma_2(j) = A001157(j).
a(n) = Sum_{i=1..n} i^2 * floor(n/i). - Enrique Pérez Herrero, Sep 15 2012
G.f.: (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 02 2017
a(n) ~ zeta(3) * n^3 / 3. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..s} (A000330(floor(n/k)) + k^2*floor(n/k)) - s*A000330(s), where s = floor(sqrt(n)). - Daniel Suteu, Nov 26 2020

A064603 Partial sums of A001158: Sum_{j=1..n} sigma_3(j).

Original entry on oeis.org

1, 10, 38, 111, 237, 489, 833, 1418, 2175, 3309, 4641, 6685, 8883, 11979, 15507, 20188, 25102, 31915, 38775, 47973, 57605, 69593, 81761, 98141, 113892, 133674, 154114, 179226, 203616, 235368, 265160, 302609, 339905, 384131, 427475, 482736

Offset: 1

Labos Elemer, Sep 24 2001

nonn

In general, Sum_{k=1..n} sigma_m(k) = Sum_{k=1..n} k^m * floor(n/k). - Daniel Suteu, Nov 08 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A001158, A064606.

Cf. A064602, A064604, A248076.

Accumulate@ Array[DivisorSigma[3, #] &, 36] (* Michael De Vlieger, Nov 03 2017 *)

a(n) = sum(j=1, n, sigma(j, 3)); \\ Michel Marcus, Nov 04 2017

a(n) = sum(k=1, n, k^3 * (n\k)); \\ Daniel Suteu, Nov 08 2018

from math import isqrt
def A064603(n): return (-(s:=isqrt(n))**3*(s+1)**2 + sum((q:=n//k)*(4*k**3+q*(q*(q+2)+1)) for k in range(1,s+1)))>>2 # Chai Wah Wu, Oct 21 2023

a(n) = a(n-1) + A001158(n) = Sum_{j=1..n} sigma_3(j), where sigma_3(j) = A001158(j).
G.f.: (1/(1 - x))*Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 23 2017
a(n) ~ Pi^4 * n^4 / 360. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..n} ((1/2) * floor(n/k) * floor(1 + n/k))^2. - Daniel Suteu, Nov 07 2018
a(n) = Sum_{k=1..n} k^3 * floor(n/k). - Daniel Suteu, Nov 08 2018

A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29

Offset: 1

Ilya Gutkovskiy, Dec 09 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
   8,  15,   37,  111,   373,   1335,  ...
  10,  21,   63,  237,   999,   4461,  ...
  14,  33,  113,  489,  2393,  12513,  ...

Index entries for sequences related to sigma(n)

Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.

Cf. A082771, A109974, A319194 (diagonal).

Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A319649_T(n,k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1,q+1) for w in range(1,s+1)))//(k+1) + int(k==0)
def A319649_gen(): # generator of terms
     return (A319649_T(k+1,n-k-1) for n in count(1) for k in range(n))
A319649_list = list(islice(A319649_gen(),30)) # Chai Wah Wu, Oct 24 2023

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} j^k*x^j/(1 - x^j).

A(n,k) = Sum_{j=1..n} sigma_k(j).

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).

cp's OEIS Frontend

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

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A064603 Partial sums of A001158: Sum_{j=1..n} sigma_3(j).

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A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

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

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