A013662 -id:A013662 - cp's OEIS frontend

A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

1, 17, 85, 273, 631, 1445, 2409, 4369, 6898, 10727, 14653, 23205, 28575, 40953, 53635, 69905, 83539, 117266, 130341, 172263, 204765, 249101, 279865, 371365, 394406, 485775, 558778, 657657, 707311, 911795, 923553, 1118481, 1245505, 1420163, 1520079, 1883154

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

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

Cf. A000583, A013662, A013663, A027848, A288415.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).

f[p_, e_] :=  (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027848(n) for odd n.
Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A343193 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 10^n.

Original entry on oeis.org

1, 9279, 92434863, 923988964495, 9239427676877311, 92393887177379735327, 923938441006918271400831, 9239384074081430755652624559, 92393840333765561759423951663423, 923938402972369921481535120722882015

Offset: 0

Karl-Heinz Hofmann, Apr 07 2021

nonn

			(1,2,2,3) is counted, but (2,4,4,6) is not, because gcd = 2.
For n=1, the size of the division tesseract matrix is 10 X 10 X 10 X 10:
.
              o------------x(w=10)------------o
             /|.                            ./ |
            / |.                           ./  |
           /  |.                          ./   |
          /   |.                         ./    |
         /    |.                      z(w=10)  |
        /     |.                      . /      |
       /      |.                     . /       |
      /       |.                   .  /     y(w=10)
     o------------------------------.o         |
    |\        /|¯¯¯¯¯¯x(w=1)¯¯¯¯¯¯/. |         |
    | w      / |                 /.| |         |
    |  \ z(w=1)|                /. | |         |
    |   \  /   |y(w=1)         /.  | |         |
    |    \/-------------------/.   | |         |
    |     |                   |    | |         |        w | sums
    |     |  Cube at w = 1    |    | |         |      ----+-----
    |     |   10 X 10 X 10    | _ _| |---------o        1 | 1000
    |     |    contains       |    / |         /        2 |  875
    |     |      1000         |   /  |        /         3 |  973
    |     |    completely     |  /   |       /          4 |  875
    |     | reduced fractions | /    |      /           5 |  992
    |     |                   |/     |     /            6 |  849
    |     /------------------- \     |    /             7 |  999
    |    /                      \    |   /              8 |  875
    |   w                        w   |  /               9 |  973
    |  /                          \  | /               10 |  868
    | /                            \ |/               ----+-----
    o -------------------------------o       sum for a(1) | 9279

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Chai Wah Wu, Table of n, a(n) for n = 0..15

Cf. A215267, A013662, A082540, A342841 (3D), A342586 (2D).
Related counts of k-tuples:
pairs: A018805, A342632, A342586;
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

from labmath import mobius
def A343193(n): return sum(mobius(k)*(10**n//k)**4 for k in range(1, 10**n+1))

Lim_{n->infinity} a(n)/10^(4*n) = 1/zeta(4) = A215267 = 90/Pi^4.

a(n) = A082540(10^n).

Edited by N. J. A. Sloane, Jun 13 2021

A347214 Decimal expansion of Sum_{k=2..4} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			3.9293142037189589133881570246986430827586871454660...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662.

Cf. A347213, A347215, A347216, A347217, A347218, A347219, A347220.

RealDigits[Total[Zeta[{2, 3, 4}]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

zeta(2)+zeta(3)+zeta(4) \\ Michel Marcus, Aug 24 2021

A347215 Decimal expansion of Sum_{k=2..5} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			4.9662419588623288397195225111556772508157680...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663.

Cf. A347213, A347214, A347216, A347217, A347218, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 5]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347216 Decimal expansion of Sum_{k=2..6} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			5.9835850208467779794340404409465977787175855550...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664.

Cf. A347213, A347214, A347215, A347217, A347218, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 6]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347217 Decimal expansion of Sum_{k=2..7} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			6.9919342982287008062738379907963945383174491155660...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665.

Cf. A347213, A347214, A347215, A347216, A347218, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 7]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347218 Decimal expansion of Sum_{k=2..8} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			7.99601165442664514565252322930504700357640...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666.

Cf. A347213, A347214, A347215, A347216, A347217, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 8]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

Equals A347217 + A013666. - R. J. Mathar, May 27 2024

A347219 Decimal expansion of Sum_{k=2..9} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			8.9980200472527273600703759985374590640620...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347220.

RealDigits[Total[Zeta[Range[2, 9]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			9.9990146223805454454075219574377780810680...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667, A013668.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347219.

RealDigits[Total[Zeta[Range[2, 10]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A363607 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^4.

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 35, 60, 85, 130, 165, 245, 286, 399, 466, 620, 680, 921, 969, 1274, 1366, 1705, 1771, 2325, 2310, 2886, 3010, 3679, 3654, 4666, 4495, 5580, 5622, 6664, 6590, 8285, 7770, 9405, 9426, 11210, 10660, 13230, 12341, 14953, 14740, 16951, 16215, 20181

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013662, A069153, A363608.
Cf. A059358, A363604, A363611.
Cf. A000203, A001157, A001158.

a[n_] := DivisorSum[n, Binomial[#, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - 3*sigma(f, 2) + 2*sigma(f)) / 6; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k,3) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,3).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_3(n) - 3*sigma_2(n) + 2*sigma_1(n)) / 6.
Dirichlet g.f.: zeta(s) * (zeta(s-3) - 3*zeta(s-2) + 2*zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

cp's OEIS Frontend

A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

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A343193 Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 10^n.

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A347214 Decimal expansion of Sum_{k=2..4} zeta(k).

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A347215 Decimal expansion of Sum_{k=2..5} zeta(k).

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A347216 Decimal expansion of Sum_{k=2..6} zeta(k).

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A347217 Decimal expansion of Sum_{k=2..7} zeta(k).

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A347218 Decimal expansion of Sum_{k=2..8} zeta(k).

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A347219 Decimal expansion of Sum_{k=2..9} zeta(k).

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