A330595 -id:A330595 - cp's OEIS frontend

A160892 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 4.

15, 105, 195, 420, 465, 1365, 855, 1680, 1755, 3255, 1995, 5460, 2745, 5985, 6045, 6720, 4605, 12285, 5715, 13020, 11115, 13965, 8295, 21840, 11625, 19215, 15795, 23940, 13065, 42315, 14895, 26880, 25935, 32235, 26505, 49140, 21105, 40005, 35685, 52080, 25845

Offset: 1

N. J. A. Sloane, Nov 19 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

Cf. A000010, A002117, A013661, A160889, A330595.

f[p_, e_] := (p^2 + p + 1)*p^(2*e - 2); a[1] = 15; a[n_] := 15*Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 15 * prod(i = 1, #f~, (f[i,1]^2 + f[i,1] + 1)*f[i,1]^(2*f[i,2] - 2));} \\ Amiram Eldar, Nov 08 2022

a(n) = 15*A160889(n). - R. J. Mathar, Feb 07 2011
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^3, where c = 5 * Product_{p prime} (1 + 1/p^2 + 1/p^3) = 5 * A330595 = 8.7446649892... .
Sum_{k>=1} 1/a(k) = (zeta(2)*zeta(3)/15) * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 0.09339604419... . (End)

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

			1.25361794500723589001447178...

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Mathematica
```
RealDigits[N[3465/2764, 105]][[1]]
```

default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.

cp's OEIS Frontend

A160892 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 4.

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A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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cp's OEIS Frontend

A160892 a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A160892 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 4.