A183699 -id:A183699 - cp's OEIS frontend

A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

1, 5, 10, 5, 26, 50, 50, 69, 10, 130, 122, 50, 170, 250, 260, 69, 290, 50, 362, 130, 500, 610, 530, 690, 26, 850, 739, 250, 842, 1300, 962, 1093, 1220, 1450, 1300, 50, 1370, 1810, 1700, 1794, 1682, 2500, 1850, 610, 260, 2650, 2210, 690, 50, 130, 2900, 850, 2810

Offset: 1

Amiram Eldar, Jul 11 2024

The number of divisors of n that are exponentially odd is A322483(n) and their sum is A033634(n).

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

Cf. A001157, A005117, A033634, A322483, A351265, A358346, A374458, A374538.

Cf. A002117, A013661, A065464, A183699.

f[p_, e_] := 1 + p^2 * (p^(4*Floor[(e-1)/2]+4) - 1) / (p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, 1 + p[i]^2 * (p[i]^(4*((e[i]-1)\2)+4) - 1) / (p[i]^4 - 1));}

a(n) = A001157(n) if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 1 + p^2 * (p^(4*floor((e-1)/2)+4) - 1) / (p^4 - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4)).
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^3) = A183699 * A065464 = 0.84677961058798544766... .

A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172

Offset: 1

Amiram Eldar, Jul 11 2024

The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

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

Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

Cf. A002117, A013661, A183699.

f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2,  f[i, 1]^(2*f[i, 2]), 0));}

a(n) = A034676(A350389(n)).
a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*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^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

A375901 Decimal expansion of the triple integral of {x/y}{y/z}{z/x} over the unit cube.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Sep 01 2024

			0.095850174913379525678536198596335370099479485204923539814301707481613569555....

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), p. viii.

Cf. A002117, A013661, A183699.

RealDigits[1 + (Zeta[3]/6 - 3/4) * Zeta[2], 10, 120, -1][[1]] (* Amiram Eldar, Sep 03 2024 *)

PARI
```
1-.75*zeta(2)+zeta(2)*zeta(3)/6
```

Integral_{0..1} Integral_{0..1} Integral_{0..1} {x/y}*{y/z}*{z/x} dx dy dz, where {w} is the fractional part of w.

Vălean shows that this is equal to 1 - 3/4 * zeta(2) + 1/6 * zeta(2) * zeta(3).

A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

Original entry on oeis.org

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

A364490 Decimal expansion of zeta(3) * primezeta(2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 26 2023

			0.543627133196479429780255713473283428069364804957661398719160636...

Eric Weisstein's World of Mathematics, Apéry's Constant.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A002117, A008472, A085548, A183699, A364488.

RealDigits[Zeta[3] PrimeZetaP[2], 10, 105][[1]]

zeta(3) * sumeulerrat(1/p, 2) \\ Amiram Eldar, Jul 28 2023

Equals Sum_{k>=1} sopf(k) / k^3, where sopf(k) is the sum of the distinct primes dividing k (A008472).

A383289 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}{y/z}{z/x})^2 dx dy dz, where {w} is the fractional part of w.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 26 2025

			0.02340961823158087268020093855006980675840442582714...

Cornel Ioan Vălean, Problem 11902, Problems and Solutions, The American Mathematical Monthly, Vol. 123, No. 4 (2016), p. 399; A Row of Zetas, Solution to Problem 11902 by Rituraj Nandan, ibid., Vol. 125, No. 2 (2018), pp. 182-184.
Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), p. viii.

Cf. A002117, A013661, A013662, A013664, A183699, A183700.

Cf. A375901 (m=1), this constant (m=2), A386564 (m=3).

RealDigits[1 - Zeta[2]/2 - Zeta[3]/2 + 7*Zeta[6]/48 + Zeta[2]*Zeta[3]/18 + Zeta[3]^2/18 + Zeta[3]*Zeta[4]/12, 10, 120, -1][[1]]
RealDigits[With[{m = 2}, 1 - 3*Sum[Zeta[j + 1], {j, 1, m}]/(2*(m + 1)) + Sum[Zeta[j + 1], {j, 1, m}] * Sum[(j + 1)*Zeta[j + 2], {j, 1, m}]/((m + 1)^2*(m + 2))], 10, 106][[1]] (* Vaclav Kotesovec, Jul 26 2025, following the general formula found by the solvers *)

1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12

Equals 1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12.

In general, Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^m dx dy dz = 1 - 3*Sum_{j=1..m} zeta(j+1)/(2*(m+1)) + (Sum_{j=1..m} zeta(j+1))*(Sum_{j=1..m} (j+1)*zeta(j+2))/((m+1)^2*(m+2)).

A386564 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}{y/z}{z/x})^3 dx dy dz, where {w} is the fractional part of w.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 26 2025

			0.00778895508409665205428360965992714119017196489266...

Cornel Ioan Vălean, Problem 11902, Problems and Solutions, The American Mathematical Monthly, Vol. 123, No. 4 (2016), p. 399; A Row of Zetas, Solution to Problem 11902 by Rituraj Nandan, ibid., Vol. 125, No. 2 (2018), pp. 182-184.
Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), section 1.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power, p. 31.

Cf. A002117, A013661, A013662, A013663, A013664, A013666, A183699, A183700.

Cf. A375901 (m=1), A383289 (m=2), this constant (m=3).

RealDigits[1 - 3*(Zeta[2]+Zeta[3]+Zeta[4])/8 + 21*Zeta[6]/320 + 7*Zeta[8]/160 + Zeta[3]^2/40 + Zeta[2]*Zeta[3]/40 + Zeta[2]*Zeta[5]/20 + Zeta[3]*Zeta[4]/16 + Zeta[3]*Zeta[5]/20 + Zeta[4]*Zeta[5]/20, 10, 120, -1][[1]]

1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20

Equal 1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20.

In general, Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^m dx dy dz = 1 - 3*Sum_{j=1..m} zeta(j+1)/(2*(m+1)) + (Sum_{j=1..m} zeta(j+1))*(Sum_{j=1..m} (j+1)*zeta(j+2))/((m+1)^2*(m+2)).

A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jun 13 2024

			0.69010488251022497818773002567827532644066623131...

Steve Fan, The shifted prime-divisor function over shifted primes, arXiv:2406.05217 [math.NT], 2024. See p. 34.

Cf. A002117, A013661, A013664, A082695, A152416, A157289, A157292, A183699.

RealDigits[(2-Zeta[2])Zeta[2]Zeta[3]/Zeta[6],10,100][[1]]

cp's OEIS Frontend

A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

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A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

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A375901 Decimal expansion of the triple integral of {x/y}{y/z}{z/x} over the unit cube.

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A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

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A364490 Decimal expansion of zeta(3) * primezeta(2).

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A383289 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^2 dx dy dz, where {w} is the fractional part of w.

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A386564 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^3 dx dy dz, where {w} is the fractional part of w.

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A373702 Decimal expansion of (2 - zeta(2))*zeta(2)*zeta(3)/zeta(6).

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A383289 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}{y/z}{z/x})^2 dx dy dz, where {w} is the fractional part of w.

A386564 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}{y/z}{z/x})^3 dx dy dz, where {w} is the fractional part of w.

A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).