A065484 -id:A065484 - cp's OEIS frontend

A101992 Numerator of Sum_{i=2..n} (-1)^i/(i*phi(i)).

1, 1, 11, 49, 59, 131, 559, 14533, 15289, 33031, 34417, 441877, 452173, 2224829, 9034451, 152504587, 155227307, 2932982513, 2967901397, 2945730677, 2971126229, 6189267977, 6250111487, 155668689479, 156604743479, 1404034379311, 1411857116311, 5835711932717

Offset: 2

Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004

I conjecture that there exists a limit for Sum_{i>=2} (-1)^i/(i*phi(i)) which is ca. 0.558.

			a(4) = 11 because Sum_{i=2..4} (-1)^i/(i*phi(i)) = 1/2 - 1/6 + 1/8 = 11/24, and the numerator of 11/24 is 11.

Amiram Eldar, Table of n, a(n) for n = 2..2296
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010 (phi), A002618, A065484.

(* Generating the sum : *) f[n_Integer]/; n >= 2 := Sum[(-1)^i/(i*EulerPhi[i]), {i, 2, n}]; (* Getting the numerator: *) a[n_Integer]/; n >=2 := Numerator[f[n]]; (* Generating the sequence : *) Table[a[n], {n, 2, 20}]
Accumulate[Table[(-1)^n/(n EulerPhi[n]),{n,2,30}]]//Numerator (* Harvey P. Dale, Mar 19 2023 *)

a(n) = numerator( Sum_{i=2..n} (-1)^i/(i*phi(i)) ).

Sum_{i>=2} (-1)^i/(i*phi(i)) = 1 - (1/5) * A065484 = 0.5592286807... . - Amiram Eldar, Nov 21 2022

More terms from Amiram Eldar, Jul 13 2019

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

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*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

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

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A345052 a(n) = A003557(n) * A048250(n) * A173557(n).

Original entry on oeis.org

1, 3, 8, 6, 24, 24, 48, 12, 24, 72, 120, 48, 168, 144, 192, 24, 288, 72, 360, 144, 384, 360, 528, 96, 120, 504, 72, 288, 840, 576, 960, 48, 960, 864, 1152, 144, 1368, 1080, 1344, 288, 1680, 1152, 1848, 720, 576, 1584, 2208, 192, 336, 360, 2304, 1008, 2808, 216, 2880, 576, 2880, 2520, 3480, 1152, 3720, 2880, 1152, 96

Offset: 1

Antti Karttunen, Jun 08 2021

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

Cf. A000010, A003557, A007434, A048250, A065484, A173557, A185197.

f[p_, e_] := (p^2 - 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 16 2022 *)

A345052(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i,1]^2)-1)*(f[i,1]^(f[i, 2]-1))); };

Multiplicative with a(p^e) = (p^2 - 1) * p^(e-1).
a(n) = A007434(n) / A003557(n) = A003557(n) * A048250(n) * A173557(n).
From Amiram Eldar, Oct 16 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/Pi^2 = 0.202642... (A185197).
Sum_{n>=1} 1/a(n) = A065484.
a(n) = A000010(n) * A048250(n). (End)

A077387 Continued fraction expansion of Product_{p prime} (1 + p/((p-1)^2*(p+1))).

Original entry on oeis.org

2, 4, 1, 9, 1, 1, 2, 1, 48, 1, 12, 1, 1, 2, 1, 1, 3, 9, 3, 3, 1, 16, 3, 1, 1, 6, 12, 50, 23, 8, 1, 1, 2, 1, 1, 1, 2, 1, 11, 1, 2, 2, 1, 1, 10, 5, 7, 2, 1, 3, 1, 4, 3, 8, 1, 2, 1, 1, 4, 11, 1, 1, 1, 1, 1, 16, 1, 1, 1, 7, 1, 13, 1, 3, 1, 6, 2, 7, 1, 2, 1, 11, 2, 1, 5, 1, 9, 4, 2, 9, 26, 1, 2, 1, 20, 1, 1, 4

Offset: 0

N. J. A. Sloane, Nov 30 2002

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A065484 (decimal expansion).

contfrac(prodeulerrat(1 + p/((p-1)^2*(p+1)))) \\ Amiram Eldar, Jun 13 2021

Extended by Ray Chandler, Sep 27 2006

Offset changed by Andrew Howroyd, Jul 06 2024

A137148 a(n) = k*phi(k), where k is the n-th nonprime number.

Original entry on oeis.org

1, 8, 12, 32, 54, 40, 48, 84, 120, 128, 108, 160, 252, 220, 192, 500, 312, 486, 336, 240, 512, 660, 544, 840, 432, 684, 936, 640, 504, 880, 1080, 1012, 768, 2058, 1000, 1632, 1248, 972, 2200, 1344, 2052, 1624, 960, 1860, 2268, 2048, 3120, 1320, 2176, 3036

Offset: 1

Artur Jasinski, Jan 23 2008

Numbers that occur in A002618 but not in A036689.

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

Cf. A002618, A018252, A036689, A065484, A136141.

a = {}; Do[If[!PrimeQ[n], AppendTo[a, n EulerPhi[n]]], {n, 1, 100}]; a

a(n) = A002618(A018252(n)). - R. J. Mathar, Jan 18 2021

Sum_{n>=1} 1/a(n) = A065484 - A136141 = 1.430699927388... . - Amiram Eldar, Oct 26 2024

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.

A370746 Decimal expansion of Sum_{k>=1} 1/(kphi(2k)), where phi is the Euler totient function (A000010).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 29 2024

The constant h in Heath-Brown et al. (2005). The asymptotic number of integers n below x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of A_{n-1} in A_n (where A_n is the simple alternating group), is ~ h*x/log(x).

The constant appears in the asymptotic formula for the count of terms of A370745.

			1.76308527715028783029826265318407173005373855503028...

D. R. Heath-Brown, Cheryl E. Praeger and Aner Shalev, Permutation groups, simple groups, and sieve methods, Isr. J. Math., Vol. 148 (2005), pp. 347-375; alternative link.
Cheryl E. Praeger, Using the finite simple groups, Gazette of the Australian Mathematical Society, Vol. 38, No. 2 (2011), pp. 93-97.
Cheryl E. Praeger, Using the finite simple groups, Eureka, Vol. 61 (2011), pp. 64-67 (reprint of the Gazette paper).
Cheryl E. Praeger, Using the finite simple groups, Asia Pacific Mathematics Newsletter, Vol. 1, No. 3 (2011), pp. 7-10 (reprint of the Gazette paper).

Cf. A000010, A065484, A370745.

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{1, 1, -2, 0, 1}, {0, 2, 3, 6, 5}, m]; RealDigits[(4/5)*Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

(4/5)* prodeulerrat(1 + p/((p-1)^2*(p+1)))

Equals (4/5)* Product_{p prime} (1 + p/((p-1)^2*(p+1))) = (4/5) * A065484.

cp's OEIS Frontend

A101992 Numerator of Sum_{i=2..n} (-1)^i/(i*phi(i)).

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A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A345052 a(n) = A003557(n) * A048250(n) * A173557(n).

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A077387 Continued fraction expansion of Product_{p prime} (1 + p/((p-1)^2*(p+1))).

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Programs

PARI

Extensions

A137148 a(n) = k*phi(k), where k is the n-th nonprime number.

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Programs

Mathematica

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.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A370746 Decimal expansion of Sum_{k>=1} 1/(k*phi(2*k)), where phi is the Euler totient function (A000010).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370746 Decimal expansion of Sum_{k>=1} 1/(kphi(2k)), where phi is the Euler totient function (A000010).