A157292 -id:A157292 - cp's OEIS frontend

A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

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

Offset: 1

Benoit Cloitre, Apr 12 2003

Equals the Dirichlet zeta-function Sum_{n>=1} A001615(n)/n^s at s=3. - R. J. Mathar, Apr 02 2011
Dressler shows that this is the average value of A014197, that is, the number of values m such that phi(m) <= n is asymptotically n times this constant. Erdős had shown earlier that this limit exists. - Charles R Greathouse IV, Nov 26 2013
From Stanislav Sykora, Nov 14 2014: (Start)
Equals lim_{n->infinity} (Sum_{k=1..n} k/phi(k))/n, i.e., the limit mean value of k/phi(k), where phi(k) is Euler's totient function.
Also equals lim_{n->infinity} (Sum_{k=1..n} 1/phi(k))/log(n).
Proofs are trivial using the formulas for Sum_{k=1..n} k/phi(k) and Sum_{k=1..n} 1/phi(k) listed in the Wikipedia link.
For the limit mean value of phi(k)/k, see A059956. (End)
The asymptotic mean of A005361. - Amiram Eldar, Apr 13 2020

			1.94359643682075920505707036257476343718785850176780571602663568890 ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.7, p. 116.
Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Paul T. Bateman, The distribution of values of the Euler function, Acta Arithmetica 21:1 (1972), pp. 329-345.
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371-378.
Paul Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.
J. von zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.
S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
D. Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
Eric Weisstein's World of Mathematics, Powerful Number.
Wikipedia, Euler's totient function.

Cf. A000010, A001615, A001694, A002117, A005117, A005361, A013661, A013664, A014197, A059956, A068468, A070243, A082696 (continued fraction), A157292.

First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]
RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* Robert G. Wilson v, Aug 11 2014 *)

PARI
```
zeta(3)*315/2/Pi^4
```

Decimal expansion of Product_{p prime} (1+1/p/(p-1)) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707...
The sum of the reciprocals of the powerful numbers, A001694. - T. D. Noe, May 03 2006
Equals A013661 * A002117 / A013664 = 1 / A068468 = zeta(3) * 315/(2*Pi^4) = zeta(3) * A157292.
Equals Sum_{k>=1} mu(k)^2/(k*phi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Euler totient function values, A000010). - Amiram Eldar, Aug 18 2020

New definition from Eric W. Weisstein, May 04 2006

A365298 a(n) is the smallest number k such that k*n is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 2, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 18, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364

Offset: 1

Amiram Eldar, Aug 31 2023

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

Cf. A157292, A335988, A356192.

f[p_, e_] := p^If[OddQ[e], Max[e, 3] - e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i, 2]%2, max(f[i, 2], 3) - f[i,2], 1))};

Multiplicative with a(p) = p^2, a(p^e) = p if e is even, and a(p^e) = 1 is e is odd and > 1.
a(n) = A356192(n)/n.
a(n) = 1 if and only if n is in A335988.
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(3*s-2) - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (2*Pi^4/315) * Product_{p prime} (1 - p^2 - p^3 + p^4 + p^8 + p^9)/(p^8*(p+1)) = 0.207915752545... .

A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2023

The abundancy index of a positive integer k is A000203(k)/k = A017665(k)/A017666(k).

The asymptotic mean of the abundancy index over all the positive integers is lim_{m->oo} (1/m) * Sum_{k=1..m} A000203(k)/k = Pi^2/6 = zeta(2) = 1.644934... (A013661).

			2.14968690301526765128219042105109416145987653275100999873...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A001694, A017665, A017666, A180114.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A240976 (squares), A245058 (even), A306633 (squarefree), A362985 (cubefull).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -3, 4, -6, 7, -7, 7, -6, 5, -3, 2, -1}, {0, 0, 0, 4, 5, 6, 0, -12, -9, -5, 0, 22}, m]; RealDigits[(2^4 + 2^2 + 2^(3/2) - 1)/(2^4 - 2)*(3^4 + 3^2 + 3^(3/2) - 1)/(3^4 - 3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

prodeulerrat((p^8 + p^4 + p^3 - 1)/(p^8 - p^2), 1/2)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A180114(k)/A001694(k).

Equals Product_{p prime} (p^4 + p^2 + p^(3/2) - 1)/(p^4 - p) = Product_{p prime} (1 + (p^2 + p^(3/2) + p - 1)/(p^4 - p)) (Jakimczuk and Lalín, 2022).

A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2023

			2.48217919642235952546167643674687698536368940971930468354...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A036966, A017665, A017666, A362986.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A245058 (even), A240976 (squares), A306633 (squarefree), A362984 (powerful).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -1, -2, 3, -2, -1, 3, -2, -2, 3, -1, -2, 3, -1, -1, 1}, {0, 0, 0, -4, 0, 6, 7, 4, 9, 0, -11, -22, -26, -21, -15, 20}, m]; RealDigits[((2^5 + 2^(10/3) + 2^3 + 2^(8/3) - 1)/(2^(10/3)*(2^(5/3) + 2^(1/3) + 1)))*((3^5 + 3^(10/3) + 3^3 + 3^(8/3) - 1)/(3^(10/3)*(3^(5/3) + 3^(1/3) + 1))) * Zeta[4/3] * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/3] - 1/2^(n/3) - 1/3^(n/3))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

zeta(4/3) * prodeulerrat((p^15 + p^10 + p^9 + p^8 - 1)/(p^10 * (p^5 + p + 1)), 1/3)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A362986(k)/A036966(k).

Equals zeta(4/3) * Product_{p prime} ((p^5 + p^(10/3) + p^3 + p^(8/3) - 1)/(p^(10/3) * (p^(5/3) + p^(1/3) + 1))).

A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

Original entry on oeis.org

1, 47, 323, 1744, 3749, 15181, 19207, 59648, 84969, 176203, 175691, 563312, 399853, 902729, 1210927, 1970176, 1503377, 3993543, 2606419, 6538256, 6203861, 8257477, 6716183, 19266304, 12105625, 18793091, 21172347, 33497008, 21218429, 56913569, 29552671, 64028672

Offset: 1

Seiichi Manyama, May 17 2024

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

Cf. A343498, A372926, A372929, A372931.
Cf. A000203, A008683.
Cf. A013661, A013664, A157292.

f[p_, e_] := p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^4*sigma(d));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^4.
a(n) = Sum_{d|n} mu(n/d) * d^4 * sigma(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, c = zeta(2)/zeta(6) = 315/(2*Pi^4) = 1.616892... (A157292). (End)
Mobius transformation of A280022. - R. J. Mathar, Jul 14 2025

A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

Original entry on oeis.org

1, 17, 163, 529, 2501, 2771, 14407, 16913, 39529, 42517, 146411, 86227, 342733, 244919, 407663, 541201, 1336337, 671993, 2345779, 1323029, 2348341, 2488987, 6156503, 2756819, 7815001, 5826461, 9605467, 7621303, 19803869, 6930271, 27705631, 17318417, 23864993

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A057660, A068963, A068970.
Cf. A001160, A008683.
Cf. A372966.
Cf. A013661, A013664, A157292.

f[p_, e_] := (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 5));

PARI
```
a(n) = sumdiv(n, d, eulerphi(d^5));
```

a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_5(d).
a(n) = Sum_{d|n} d^(5-m) * phi(d^m) for m > 0.
G.f.: Sum_{k>=1} k^(5-m) * phi(k^m) * x^k/(1 - x^k) for m > 0.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(2) = 2*Pi^4/315 = 0.6184704192... (1/A157292). (End)

A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 99, 84, 144

Offset: 1

Amiram Eldar, Aug 25 2023

The number of these divisors is A252505(n).

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

Cf. A000203, A000290, A005117, A034448, A046100, A157292, A252505.

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

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

Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = (1 + p)*(1 + p^(e - 1)) if e >= 3.
a(n) >= A034448(n), with equality if and only if n is squarefree number (A005117).
a(n) <= A000203(n), with equality if and only if n is biquadratefree number (A046100).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 315/(4*Pi^4) = A157292 / 2 = 0.808446... .

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

A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

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A365298 a(n) is the smallest number k such that k*n is a cubefull exponentially odd number (A335988).

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A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

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A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

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A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

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A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

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A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.

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