A091724 -id:A091724 - cp's OEIS frontend

A228127 Numbers n such that first n digits of A091724 (Decimal expansion of E^(2*EulerGamma)) gives a prime number.

1, 2, 3, 1080

Offset: 1

Robert Price, Aug 26 2013

a(5) > 10^4.

Eric Weisstein's World of Mathematics, ConstantPrimes
Eric Weisstein's World of Mathematics, IntegerSequencePrimes

nn=1100;With[{c=RealDigits[E^(2*EulerGamma),10,nn][[1]]},Select[ Range[ nn], PrimeQ[FromDigits[Take[c,#]]]&]] (* Harvey P. Dale, Nov 13 2014 *)

A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

Original entry on oeis.org

1, 1, 3, 1, 7, 3, 3, 7, 1, 7, 9, 7, 14, 3, 15, 1, 31, 1, 39, 5, 7, 3, 9, 15, 7, 7, 39, 7, 7, 5, 9, 31, 21, 31, 15, 7, 91, 39, 5, 31, 15, 7, 24, 7, 5, 3, 9, 31, 8, 7, 21, 10, 49, 39, 15, 15, 91, 7, 45, 31, 28, 9, 91, 1, 31, 7, 9, 7, 21, 5, 6, 5, 65, 91, 3, 91, 21

Offset: 1

Amiram Eldar, Feb 27 2024

			Fractions begin with: 1, 1/2, 3/2, 1/2, 7/2, 3/4, 3, 7/8, 1, 7/6, 9/2, 7/12, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930, A289336, A073858 (positions of 1's), A289412, A370690 (denominators).

Cf. A080130, A091724.

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Numerator

a(n) = {my(f = factor(n)); numerator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

Let f(n) = a(n)/A370690(n) = A062402(n)/A062401(n).
Formulas from De Koninck and Luca (2007):
lim sup_{n->oo} f(n)/log_2(n)^2 = exp(2*gamma) (A091724).
lim inf_{n->oo} f(n)/log_2(n)^2 = delta exists, and exp(-gamma)/40 <= delta <= 2*exp(-gamma).
Sum_{k=1..n} f(k) = c * exp(2*gamma) * log_3(n)^2 * n + O(n * log_3(n)^(3/2)), where c = Product_{p prime} (1 - 3/(p*(p + 1)) + 1/(p^2*(p + 1)) + ((p-1)^3/p^2)*Sum_{k>=3} 1/(p^k-1)) = 0.45782563109026414241... .

A078559 Numerator of Product_{i=1..n} (p_i + 1)/(p_i - 1) where p_i is the i-th prime.

Original entry on oeis.org

3, 6, 9, 12, 72, 84, 189, 21, 252, 270, 288, 304, 1596, 152, 3648, 49248, 295488, 1526688, 17302464, 622888704, 640191168, 1707176448, 10243058688, 23046882048, 23527025424, 599939148312, 47054050848, 2540918745792

Offset: 1

Labos Elemer, Dec 06 2002

R. K. Guy, Unsolved Problems in Number Theory, B48.

Robert Israel, Table of n, a(n) for n = 1..1701 (b-file corrected by _Georg Fischer_, Jan 16 2019)
F. Smarandache, Only Problems, Not Solutions!.

Cf. A000203, A002110, A000005, A005867, A054640, A020492, A078558, A091724, A013661.

Denominators are in A078560.

Q:= 1: p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  Q:= Q * (p+1)/(p-1);
  A[n]:= numer(Q);
od:
seq(A[i],i=1..100); # Robert Israel, May 11 2018

Numerator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* Alonso del Arte, Aug 23 2011 *)

a(n) = numerator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ Michel Marcus, May 11 2018

a(n) = A054640(n)/A078558(n).

a(n)/A078560(n) ~ C*log^2(prime(n)), where C = exp(2*gamma)/zeta(2) = 6(e^gamma/pi)^2 = A091724 / A013661. Physics note: (a(n)/A078560(n) - 1)/(a(n)/A078560(n) + 1) = tanh(Sum_{k=1..n} arctanh(1/prime(k))) is the relativistic sum of n velocities c/2, c/3, ..., c/prime(n), in units where the speed of light c = 1. - Thomas Ordowski, Nov 06 2024

Improved definition from Franklin T. Adams-Watters, Dec 02 2005

A246061 Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 11 2014

			1.201303559967362241247555959207383482453838449427113...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

Eric Weisstein's MathWorld, Twin Primes Constant.

Cf. A005597, A091724.

digits = 101; s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, digits + 60}]; RealDigits[Exp[2*EulerGamma]/(4*C2), 10, digits] // First

exp(2*Euler)/(4*prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Apr 27 2025

Equals exp(2*EulerGamma)/(4*C_2), where C_2 is the twin primes constant A005597.

A273556 Decimal expansion of Rosser's constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 25 2016

Named after the American logician and mathematician John Barkley Rosser, Sr. (1907-1989). - Amiram Eldar, Jun 20 2021

			0.832429065661945278030805943531465575045445318077417053240893991296...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), pp. 64-94, eq. (2.14).
Eric Weisstein's World of Mathematics, Twin Primes Constant.

Cf. A005597, A091724, A246061.

digits = 98; s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n) )*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 60}];
RealDigits[4*C2/Exp[2*EulerGamma], 10, digits] // First

4 * exp(-2*Euler) * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 17 2021

4*C_2/exp(2*EulerGamma), where C_2 is the twin primes constant.

Equals lim_{x->inf} Product_{2 < p <= x} (1-2/p)*log(x)^2.

cp's OEIS Frontend

A228127 Numbers n such that first n digits of A091724 (Decimal expansion of E^(2*EulerGamma)) gives a prime number.

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A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

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A078559 Numerator of Product_{i=1..n} (p_i + 1)/(p_i - 1) where p_i is the i-th prime.

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A246061 Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).

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A273556 Decimal expansion of Rosser's constant.

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