A065485 -id:A065485 - cp's OEIS frontend

A078072 Continued fraction for constant defined in A065485.

2, 1, 4, 1, 3, 5, 2, 2, 2, 4, 3, 2, 1, 3, 2, 1, 1, 1, 8, 2, 2, 28, 1, 1, 4, 10, 1, 9, 62, 4, 1, 1, 1, 2, 3, 1, 3, 3, 9, 2, 1, 7, 1, 1, 1, 4, 1, 1, 2, 1, 8, 1, 5, 7, 2, 64, 1, 5, 2, 2, 2, 4, 1, 14, 1, 3, 2, 2, 2, 23, 1, 9, 1, 20, 2, 2, 12, 1, 1, 3, 1, 2, 1, 3, 7, 4, 3, 12, 1, 4, 2, 3, 1, 4, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065485 (decimal expansion).

localprec(100); contfrac(prodeulerrat(1 + 1/(p-1)^2)) \\ Michel Marcus, Jul 13 2021

Offset changed by Andrew Howroyd, Jul 06 2024

A272030 Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 25 2016

			3.17045934214256636532648824888226302856125443631798948742143398...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 117.

Cf. A001620, A005596, A065484, A065485, A080130, A082695, A082695, A083343.

digits = 98; B3 = EulerGamma - NSum[PrimeZetaP'[n], {n, 2, Infinity}, WorkingPrecision -> 2 digits, NSumTerms -> 200]; RealDigits[Log[2 Pi] + B3, 10, digits][[1]]

C = log(2*Pi) + EulerGamma - Sum_{n >= 2} P'(n), where P'(n) is the prime zeta P function derivative.

A323333 The Euler phi function values of the powerful numbers, A000010(A001694(n)).

Original entry on oeis.org

1, 2, 4, 6, 8, 20, 18, 16, 12, 42, 32, 24, 54, 40, 36, 110, 100, 64, 48, 156, 84, 80, 72, 120, 162, 128, 96, 272, 108, 294, 342, 168, 160, 144, 252, 220, 200, 256, 506, 192, 500, 216, 360, 312, 486, 336, 320, 812, 288, 240, 930, 440, 324, 400, 512, 660, 600

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Murata's constant Product_{p prime}(1 + 1/(p-1)^2) (A065485).

Sequence is injective: no value occurs more than once. - Amiram Eldar and Antti Karttunen, Sep 30 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A000010, A001694, A002618 (a subsequence), A065485, A082695, A112526, A323332.

EulerPhi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after Harvey P. Dale at A001694 *)

lista(nn) = apply(x->eulerphi(x), select(x->ispowerful(x), vector(nn, k, k))); \\ Michel Marcus, Jan 11 2019

A241194 Numerator of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 4, 1, 2, 2, 11, 6, 14, 4, 10, 12, 1, 4, 20, 5, 1, 2, 16, 26, 1, 3, 2, 24, 8, 22, 18, 4, 4, 1, 41, 21, 44, 4, 36, 1, 3, 10, 8, 12, 56, 6, 14, 48, 4, 2, 1, 65, 33, 4, 22, 12, 46, 36, 16, 12, 4, 39, 8, 2, 86, 28, 5, 89, 20, 10, 2, 95

Offset: 1

T. D. Noe, Apr 17 2014

The denominators are in A241195. The new minima of phi(p-1)/(p-1) occur at primes listed in A241196. The numerator and denominator of those terms are in A241197 and A241198.

For primes p>2, the fraction phi(p - 1)/(p - 1) has the maximum value = 1/2 if and only if p is in A019434. - Geoffrey Critzer, Dec 30 2014

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 117.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of some sequences of numbers, III., J. London Math. Soc. 13 (1938), pp. 119-127.
Imre Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), pp. 278-289.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.

Cf. A000010, A005596, A008330 (phi(prime(n)-1)), A065485, A241195, A241196, A241197, A241198.

Cf. A076512, A006093.

[Numerator(EulerPhi(NthPrime(n)-1)/(NthPrime(n)-1)): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

seq(numer(numtheory:-phi(ithprime(i)-1)/(ithprime(i)-1)), i=1..100); # Robert Israel, Jan 11 2015

Numerator[Table[EulerPhi[p - 1]/(p - 1), {p, Prime[Range[100]]}]]

lista(nn) = forprime(p=2, nn, print1(numerator(eulerphi(p-1)/(p-1)), ", ")); \\ Michel Marcus, Jan 03 2015

From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A241195(n) = 0.373955... (Artin's constant, A005596).
Asymptotic mean of inverse ratio: lim_{m->oo} (1/m) * Sum_{n=1..m} A241195(n)/a(n) = 2.826419... (Murata's constant, A065485). (End)
a(n) = A076512(A006093(n)). - Ridouane Oudra, Mar 24 2025

A005722 a(n) = (prime(n) - 1)^2.

Original entry on oeis.org

1, 4, 16, 36, 100, 144, 256, 324, 484, 784, 900, 1296, 1600, 1764, 2116, 2704, 3364, 3600, 4356, 4900, 5184, 6084, 6724, 7744, 9216, 10000, 10404, 11236, 11664, 12544, 15876, 16900, 18496, 19044, 21904, 22500, 24336, 26244, 27556, 29584, 31684, 32400, 36100

Offset: 1

Scorpion(AT)aol.com

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Richard K. Guy, Letter to N. J. A. Sloane, 1987.

Cf. A001248, A005597, A006093, A065485, A086242, A095874, A192134.

[(p-1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

A005722:=n->(ithprime(n)-1)^2; seq(A005722(n), n=1..40); # Wesley Ivan Hurt, Mar 27 2014

Table[(Prime[n] - 1)^2, {n, 40}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = (prime(n) - 1)^2; \\ Michel Marcus, Nov 09 2020

a(n) = A192134(A095874(A001248(n))) - 1. - Reinhard Zumkeller, Jun 26 2011
a(n) = A006093(n)^2. - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = A086242. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065485.
Product_{n>=2} (1 - 1/a(n)) = A005597. (End)

A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

Original entry on oeis.org

1, 2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 120, 16, 272, 36, 342, 80, 252, 220, 506, 48, 100, 312, 54, 168, 812, 240, 930, 32, 660, 544, 840, 72, 1332, 684, 936, 160, 1640, 504, 1806, 440, 360, 1012, 2162, 96, 294, 200, 1632, 624, 2756, 108, 2200, 336, 2052, 1624

Offset: 1

R. J. Mathar, Feb 19 2018

A permutation of A323333. - Amiram Eldar, Sep 19 2020

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

Cf. A000010, A007947, A065485, A173557, A323333.

A299822 := proc(n)
    local a,p,e,pe;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ; e:= pe[2] ;
        a := a*p*(p-1)*p^(e-1) ;
    end do:
    a ;
end proc:
seq(A299822(n),n=1..130) ;

Array[EulerPhi[#] SelectFirst[Reverse@ Divisors@ #, SquareFreeQ] &, 58] (* Michael De Vlieger, Feb 20 2018 *)
f[p_, e_] := (p-1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]); \\ Michel Marcus, Jun 24 2019

a(n) = A000010(n)*A007947(n) = n*A173557(n).
Dirichlet g.f.: zeta(s-1)*Product_{p prime} (1 - 2*p^(1-s) + p^(2-s)), corrected by Vaclav Kotesovec, Dec 18 2019
Multiplicative with a(p^e) = p*(p-1)*p^(e-1).
a(n) = n*abs(A023900(n)). (Trivially rephrasing a formula in A173557.) - Omar E. Pol, Feb 19 2018
a(2^e) = 2^e. (Special case of above.) - Omar E. Pol, Feb 19 2018
A003557(n) | a(n). - R. J. Mathar, Feb 26 2018
From Vaclav Kotesovec, Dec 18 2019: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(s-2) * Product_{primes p} (1 + 2*p^(3-2*s) - p^(4-2*s) - 2*p^(1-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927... (End)
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p-1)^2) = 2.826419... (A065485). - Amiram Eldar, Sep 19 2020
G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n)^2 = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 2*x^2 - 6*x^3 - 4*x^4 - 20*x^5 + 12*x^6 - 42*x^7 - 8*x^8 - 18*x^9 + 40*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022

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

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.

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

cp's OEIS Frontend

A078072 Continued fraction for constant defined in A065485.

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A272030 Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.

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A323333 The Euler phi function values of the powerful numbers, A000010(A001694(n)).

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A241194 Numerator of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

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A005722 a(n) = (prime(n) - 1)^2.

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A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

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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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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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