A183699 -id:A183699 - cp's OEIS frontend

A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

1, 4, 9, 18, 25, 36, 49, 78, 87, 100, 121, 162, 169, 196, 225, 326, 289, 348, 361, 450, 441, 484, 529, 702, 645, 676, 807, 882, 841, 900, 961, 1334, 1089, 1156, 1225, 1566, 1369, 1444, 1521, 1950, 1681, 1764, 1849, 2178, 2175, 2116, 2209, 2934, 2443, 2580

Offset: 1

Vladeta Jovovic, Jul 21 2001

If k is squarefree (cf. A005117) then A062952(k) = k^2. - Benoit Cloitre, Apr 16 2002

Inverse Möbius transform of A062354(n). - Wesley Ivan Hurt, Jul 26 2025

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

Cf. A005117, A029939, A057660, A062952.
Cf. A002117, A013661, A183699, A330523.
Cf. A062354.

f[p_, e_] := (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{d|n} phi(d)*sigma(d).
a(n) = Sum_{k=1..n} sigma(n/gcd(n, k)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A183699 * A330523 / 3. - Amiram Eldar, Oct 30 2022

A347213 Decimal expansion of zeta(2) + zeta(3).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			2.8469909700078207218721533281574751799839361...

Omran Kouba, A zeta series, Solution to Problem 1930, Mathematics Magazine, Vol. 87, No. 4 (2014), pp. 296-298.
Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A183699.

Cf. A347214, A347215, A347216, A347217, A347218, A347219, A347220.

RealDigits[Zeta[2] + Zeta[3], 10, 100][[1]] (* Amiram Eldar, Jun 09 2022 *)

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

Equals 1 + Sum_{k>=2} k * (k - Sum_{i=2..k} zeta(i)) (Kouba, 2014). - Amiram Eldar, Jun 09 2022

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

Also zetamult(2, 2, 1). - Charles R Greathouse IV, Jan 04 2017

			0.2288103976033537597687461489416887919325093427198821602294071...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Jonathan Borwein and Roland Girgensohn, Evaluation of triple Euler Sums, Elec. Jour. of Comb., Vol. 3, Issue 1, 1996. Article R23 (see page 21).
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[3*Zeta[2]*Zeta[3] - (11/2)*Zeta[5], 10, 104] // First

zetamult([3,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult([2,2,1]) \\ Charles R Greathouse IV, Jan 04 2017

Equals Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^2)) = 3*zeta(2)*zeta(3) - (11/2)*zeta(5).

A258986 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.711566197550572432096973806086402612092561204438339236492222496457686...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[(9/2)*Zeta[5] - 2*Zeta[2]*Zeta[3], 10, 103] // First

zetamult([2,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(2,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = (9/2)*zeta(5) - 2*zeta(2)*zeta(3).

Equals Sum_{i, j >= 1} 1/(i^3*j^2*binomial(i+j, i)). More generally, for n >= 2, Sum_{i, j >= 1} 1/(i^n*j^2*binomial(i+j, i)) = zeta(2)*zeta(n) - zeta(n+2) - zeta(n,2). - Peter Bala, Aug 05 2025

A183700 Decimal expansion of zeta(3)*zeta(4), the product of two Riemann zeta values.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 06 2011

Equals the Dirichlet zeta-function sum_{n>=1} A000203(n)/n^s at s=4.

			Equals 1.301014114532488574154...

Cf. A183699 (s=3).

Maple
```
evalf(Zeta(3)*Zeta(4)) ;
```

RealDigits[N[Zeta[3] * Zeta[4], 75]][[1]] (* Alonso del Arte, Jan 06 2011 *)

Equals A002117 * A013662.

A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

Original entry on oeis.org

1, 5, 13, 21, 31, 65, 57, 85, 130, 155, 133, 273, 183, 285, 403, 341, 307, 650, 381, 651, 741, 665, 553, 1105, 806, 915, 1210, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2730, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 4030, 2765, 2257, 4433, 2850, 4030

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000290 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

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

Cf. A000290, A001001, A192065, A183699.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
Array[a, 50] (* Jean-François Alcover, Jul 03 2017 *)
f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ Michel Marcus, Jul 03 2017; corrected Jun 12 2022

L.g.f.: log(Product_{k>=1} (1 + x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A001001(n) for odd n.
Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (End)

A322485 The sum of the semi-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 19, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 76, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 71, 84, 144

Offset: 1

Amiram Eldar, Dec 11 2018

A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1 (see A322483).

			The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their sum is 11, thus a(8) = 11.

J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Pentti Haukkanen, Basic properties of the bi-unitary convolution and the semi-unitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305-315.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.

Cf. A000203, A005117, A034448, A183699, A188999, A322483.

f[p_, e_] := (p^Floor[(e+1)/2] - 1)/(p-1) + p^e; susigma[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[susigma, 100]

a(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k,1], e=f[k,2]); f[k,1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Dec 14 2018

Multiplicative with a(p^e) = sigma(p^floor((e-1)/2)) + p^e = (p^floor((e+1)/2) - 1)/(p-1) + p^e.
In particular a(p) = p + 1, a(p^2) = p^2 + 1, a(p^3) = p^3 + p + 1.
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 0.7004703314... . - Amiram Eldar, Nov 24 2022

A369717 The sum of divisors of the smallest powerful number that is a multiple of n.

Original entry on oeis.org

1, 7, 13, 7, 31, 91, 57, 15, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 195, 31, 1281, 40, 399, 871, 2821, 993, 63, 1729, 2149, 1767, 91, 1407, 2667, 2379, 465, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863

Offset: 1

Amiram Eldar, Jan 30 2024

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

Cf. A000203, A001694, A197863, A369716, A369718, A369720.

Cf. A002117, A013661, A183699.

f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,2] == 1, f[i,2] = 2)); sigma(f);}

a(n) = A000203(A197863(n)).
Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
a(n) >= A000203(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .

A369718 The sum of unitary divisors of the smallest powerful number that is a multiple of n.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 9, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 90, 26, 850, 28, 250, 842, 1300, 962, 33, 1220, 1450, 1300, 50, 1370, 1810, 1700, 234, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 140

Offset: 1

Amiram Eldar, Jan 30 2024

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

Cf. A034448, A001694, A197863, A369716, A369717, A369721.

Cf. A002117, A013661, A183699.

f[p_, e_] := If[e == 1, p^2 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1 + f[i,1]^2, 1 + f[i,1]^f[i,2]));}

a(n) = A034448(A197863(n)).
Multiplicative with a(p) = p^2 + 1 and a(p^e) = p^e + 1 for e >= 2.
a(n) >= A034448(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(s-1) - 1/p^(2*s-3) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6 - 2/p^7 + 1/p^8) = 0.73644353930922037459... .

A123732 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^3.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto, Nov 18 2006

The sum over (-1)Sigma(m)/m^2 is divergent.

			1.46400354360318202593842489434542708983961995794945...

Cf. A049060, A123733, A123734, A183699.

prodeulerrat(1 + (p^3-p^2+1)/((p-1)^2*(p+1)*(p^2+p+1))) \\ Amiram Eldar, Aug 26 2022

Equals Sum_{m>=1} A049060(m)/m^3.

Equals Product_{p prime} (1 + (p^3-p^2+1)/((p-1)^2*(p+1)*(p^2+p+1))). - Amiram Eldar, Aug 26 2022

40 more digits from R. J. Mathar, Dec 19 2010

More terms from Amiram Eldar, Aug 26 2022

cp's OEIS Frontend

A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

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A347213 Decimal expansion of zeta(2) + zeta(3).

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A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

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A258986 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).

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A183700 Decimal expansion of zeta(3)*zeta(4), the product of two Riemann zeta values.

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A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

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A322485 The sum of the semi-unitary divisors of n.

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