A080130 -id:A080130 - cp's OEIS frontend

A244499 Decimal expansion of e/gamma, the ratio of Euler number and the Euler-Mascheroni constant.

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

Offset: 1

Stanislav Sykora, Jun 29 2014

			4.709300169327103330744143217754700463516616783290647196...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 1.10, page 2.

Stanislav Sykora, Table of n, a(n) for n = 1..2019
Ovidiu Furdui, Problem 1764, Mathematics Magazine, Vol. 80, No. 1 (2007), pp. 77-78; Euler-Mascheroni meets e, Solution to Problem 1764 by Edward Schmeichel, ibid., Vol. 81, No. 1 (2008), p. 67.

Cf. A001008, A001113, A001620, A002805, A073004, A080130, A244274.

R:= RealField(100); Exp(1)/EulerGamma(R); // G. C. Greubel, Aug 30 2018

RealDigits[E/EulerGamma, 10, 100][[1]] (* G. C. Greubel, Aug 30 2018 *)

PARI
```
exp(1)/Euler
```

Equals lim_{n->oo} (g(n)^gamma/gamma^g(n))^(2*n), where g(n) = H(n) - log(n) and H(n) = A001008(n)/A002805(n) is the n-th harmonic number (Furdui, 2007 and 2013). - Amiram Eldar, Mar 26 2022

A341749 Numbers k such that gcd(k, phi(k)) > log(log(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96

Offset: 1

Amiram Eldar, Feb 18 2021

nonn

First differs from A080197 at n = 28.
Erdős et al. (2008) proved that the asymptotic density of numbers k such that gcd(k, phi(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).
There are only 8 cyclic numbers (A003277) in this sequence: 1, 2, 3, 5, 7, 11, 13, 15. All the other terms are in A060679. The first term of A060679 which is not in this sequence is 1622.

			16 is a term since gcd(16, phi(16)) = gcd(16, 8) = 8 > log(log(16)) = 1.0197...
17 is not a term since gcd(17, phi(17)) = gcd(17, 16) = 1 < log(log(17)) = 1.0414...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, Florian Luca and Carl Pomerance, On the proportion of numbers coprime to a given integer, in: J.-M. De Koninck, A. Granville and F. Luca (eds.), Anatomy of Integers, AMS, 2008, pp. 47-64.
Wikipedia, Dickman function.

Cf. A000010, A001620, A003277, A009195, A060679, A080130, A227242, A341750.

Select[Range[100], GCD[#, EulerPhi[#]] > Log[Log[#]] &]

isok(k) = (k==1) || (gcd(k, eulerphi(k)) > log(log(k))); \\ Michel Marcus, Feb 19 2021

A341750 Numbers k such that gcd(k, sigma(k)) > log(log(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 20, 22, 24, 26, 28, 30, 33, 34, 38, 40, 42, 44, 45, 46, 48, 51, 52, 54, 56, 58, 60, 62, 66, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 99, 102, 104, 105, 106, 108, 110, 112

Offset: 1

Amiram Eldar, Feb 18 2021

nonn

Pollack (2011) proved that the asymptotic density of numbers k such that gcd(k, sigma(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).

There are only 10 terms of A014567 in this sequence: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13.

			15 is a term since gcd(15, sigma(15)) = gcd(15, 24) = 3 > log(log(15)) = 0.996...
16 is not a term since gcd(16, sigma(16)) = gcd(16, 31) = 1 < log(log(16)) = 1.0197...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J., Vol. 60, No. 1 (2011), pp. 199-214.
Wikipedia, Dickman function.

Cf. A000203, A001620, A014567, A080130, A009194, A227242, A341749.

Select[Range[100], GCD[#, DivisorSigma[1, #]] > Log[Log[#]] &]

isok(k) = (k==1) || (gcd(k, sigma(k)) > log(log(k))); \\ Michel Marcus, Feb 20 2021

A125313 Decimal expansion of 2*exp(-gamma).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Dec 08 2006

			1.12291896713377033964828642958176157353142077385030633630831815209...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3, Landau-Ramanujan constant, p. 100.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal 1: 12-28, (1995) DOI:10.1080/03461238.1995.10413946.
Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.
Simon Plouffe, Plouffe's Inverter.
S. K. Wilson and B. R. Duffy, An asymptotic analysis of small holes in thin fluid layers, Journal of Engineering Mathematics, July 1996, Volume 30, Issue 4, pp 445-457.

R:= RealField(100); 2*Exp(-EulerGamma(R)); // G. C. Greubel, Sep 05 2018

RealDigits[2*Exp[-EulerGamma], 10, 111][[1]]

default(realprecision, 100); 2*exp(-Euler) \\ G. C. Greubel, Sep 05 2018

Equals 2*A080130, 2*A001113^(-A001620) and 2/A073004 = 2/A068985^A001620.
Equals A088540 * A088541. - Jean-François Alcover, Jun 04 2014
Equals exp(A002162 - A001620). - John W. Nicholson, Apr 03 2015

A181110 Decimal expansion of 1/zeta(2) - 1/e^gamma, where gamma is the Euler-Mascheroni constant and zeta(2) = Pi^2/6.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Oct 03 2010

Zeta(2) is A013661 and e^gamma is A073004.

Number theory use in Cellarosi et al., p. 9. Abstract: "We present a limit theorem describing the behavior of a probabilistic model for squarefree numbers. The limiting distribution has a density that comes from the Dickman-De Bruijn function and is constant on the interval [0,1]. We also provide estimates for the error term in the limit theorem."

			0.046467618287141458839133564467485...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Francesco Cellarosi, Yakov G. Sinai, Non-Standard Limit Theorems in Number Theory, arXiv:1010.0035 [math.PR], 2010.

Cf. A001620, A013661.

SetDefaultRealField(RealField(100)); R:= RealField(); 6/Pi(R)^2 - Exp(-EulerGamma(R)); // G. C. Greubel, Sep 06 2018

Join[{0}, RealDigits[1/Zeta[2] - Exp[-EulerGamma], 10, 100][[1]]] (* G. C. Greubel, Sep 06 2018 *)

1/zeta(2) - exp(-Euler) \\ Charles R Greathouse IV, Mar 10 2016

Equals A059956 - A080130.

Offset and leading zeros normalized by R. J. Mathar, Oct 05 2010

A360895 Decimal expansion of exp(exp(-gamma)) where gamma is the Euler-Mascheroni constant A001620.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Feb 25 2023

Theorem: Let p(n) be the smallest prime such that Product_{prime p<=p(n)} 1/(1-1/p) >= n. Then lim_{n->oo} p(n+1)/p(n) = exp(exp(-gamma)).
Proof. Follow Mertens's Third Theorem Product_{p<=x} 1/(1-1/p) ~ log(x)/exp(-gamma).
For any particular integer n, it follows from the equations n = log(x_n)/exp(-gamma) -> x_n = exp(n*exp(-gamma)) and n+1 = log(x_n+1)/exp(-gamma) -> x_n+1 = exp((n+1)*exp(-gamma)) that lim_{n->oo} exp((n+1)*exp(-gamma))/exp((n)*exp(-gamma)) = exp(exp(-gamma)).
Convergence table:
n   p(n)        truncated Euler product up to p(n)   ratio p(n)/p(n-1)
42  17427088769 42.0000000010939727723681242652955   1.7532416978341651
43  30553756811 43.0000000012946363551468233325186   1.7532335558736718
44  53567706007 44.0000000002803554088007272169139   1.7532281329055578
45  93916601047 45.0000000002681963271546340553884   1.7532317145469581
46 164657625967 46.0000000002470257389410099668348   1.7532323799133028
47 288682860119 47.0000000001305074313442036255929   1.7532310357544971
48 506127311983 48.0000000000705764045487316221655   1.7532295189758258
oo  oo           oo                                  1.7532294434956945

			1.753229443495694582297365429644...

Wikipedia, Mertens' theorems

Cf. A001620, A080130, A061556, A091440, A167348.

RealDigits[N[Exp[Exp[-EulerGamma]], 115]][[1]]

exp(exp(-Euler)) \\ Michel Marcus, Feb 25 2023

Equals exp(A080130).
Limit_{n->oo} A091440(n+1)/A091440(n).
Limit_{n->oo} A061556(n+1)/A061556(n).
Limit_{n->oo} A167348(n+1)/A167348(n).

A049012 Composite numbers n such that number of nonprime d with 0 < d < n, gcd(n,d)=1, is pi(n).

Original entry on oeis.org

33, 75, 94, 106, 118, 1540, 2442, 5340

Offset: 1

Naohiro Nomoto

Numbers n such that 2*A000720(n) = A000010(n) + A001221(n). - Max Alekseyev, Aug 22 2013

The sequence is finite since the l.h.s. grows as 2n/log(n), while the r.h.s. is asymptotically at least A080130*n/log(log(n)). In fact, known bounds for A000720 and A000010 imply that there are no terms above 10^7, and thus the sequence is full. - Max Alekseyev, Oct 29 2019

			gcd(33,d)=1: d=1,4,8,10,14,16,20,25,26,28,32, pi(33)=11, so 33 is a term.

isok(n) = {if (isprime(n) , return (0)); nb = 0; for (d=1, n-1, if (! isprime(d) && gcd(n, d) == 1, nb++);); return (nb == primepi(n));} \\ Michel Marcus, Jul 14 2013

More terms from Michel Marcus, Jul 14 2013

Keywords fini, full added by Max Alekseyev, Oct 29 2019

A235213 Nearest prime to (2^(e^-gamma))^n, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 31, 47, 73, 107, 157, 233, 347, 509, 743, 1103, 1627, 2399, 3541, 5227, 7717, 11383, 16811, 24799, 36599, 54011, 79699, 117619, 173573, 256163, 378041, 557891, 823309, 1215017, 1793081, 2646167, 3905059, 5762969, 8504759, 12550991, 18522269

Offset: 2

Robert G. Wilson v, Jan 04 2014

The nearest integer to (2^(e^-gamma))^n is very close to A018133.

Eric W. Weisstein's World of Mathematics, Wagstaff's Conjecture
Eric W. Weisstein's World of Mathematics, Eberhart's Conjecture

Cf. A000043, A001620, A080130.

f[n_] := Block[{a = (2^Exp[-EulerGamma])^n}, Nearest[{NextPrime[a], NextPrime[a, -1]}, a][[1]]]; Array[f, 42, 2]

a(n)=my(t=2^(exp(Euler)*n),mn=precprime(t),mx=nextprime(t)); if(mx-tCharles R Greathouse IV, Jan 04 2014

Nearest prime to (2^(e^-A001620))^n = (2^A080130)^n.

A241532 Decimal expansion of 2e^(-2gamma), gamma being the Euler constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2014

See Table 1, last row, first column, of the Bailey, Borwein and Crandall link. - Wolfdieter Lang, Apr 25 2014

			0.6304735033743867961220401927108789...

D.H. Bailey, J.M. Borwein, R.E. Crandall, Integrals of the Ising class

Cf. A073004, A080130.

RealDigits[2*E^(-2*EulerGamma), 10, 100] // First

Limit_{n -> infinity} (2^n/n!)*integral_{0..infinity} p*BesselK(0,p)^n dp. - after Bailey & Borwein.

Equals 2*A080130^2.

A321161 Decimal expansion of Wilf's formula: Product_{k>=1} exp(-1/k)(1 + 1/k + 1/(2k^2)) = exp(-gamma)*cosh(Pi/2)/(Pi/2).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 11 2019

The formula was discovered by Wilf in 1997.

			0.896871242167379021690230319086367005622530649081704...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2011, p. 366.

Chao-Ping Chen and Richard B. Paris, Generalizations of two infinite product formulas, Integral Transforms and Special Functions, Vol. 25, No. 7 (2014), pp. 547-551.
Chao-Ping Chen and Richard B. Paris, On the asymptotics of products related to generalizations of the Wilf and Mortini problems, Integral Transforms and Special Functions, Vol. 27, No. 4 (2016), pp. 281-288, alternative link.
Chao-Ping Chen and Richard B. Paris, On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas, Applied Mathematics and Computation, Vol. 293 (2017), pp. 30-39, preprint, arXiv:1511.09217 [math.CA], 2015.
Junesang Choi, Jungseob Lee and H.M. Srivastava, A generalization of Wilf's formula, Kodai Mathematical Journal, Vol. 26, No. 1 (2003), pp. 44-48.
Junesang Choi and H.M. Srivastava, Integral Representations for the Euler-Mascheroni Constant gamma, Integral Transforms and Special Functions, Vol. 21, No. 9 (2010), pp. 675-690.
J. Lopez-Bonilla and R. López-Vázquez, On an Identity of Wilf for the Euler-Mascheroni's Constant, Prespacetime Journal, Vol. 9, No. 6 (2018), pp. 516-518.
Herbert S. Wilf, Problem 10588, The American Mathematical Monthly, Vol. 104, No. 5 (1997), p. 456.

Cf. A001620, A042972, A049006, A073004, A080130.

RealDigits[Exp[-EulerGamma]*Cosh[Pi/2]/(Pi/2), 10, 100][[1]]

exp(-Euler)*cosh(Pi/2)/(Pi/2) \\ Michel Marcus, Jan 15 2019

cp's OEIS Frontend

A244499 Decimal expansion of e/gamma, the ratio of Euler number and the Euler-Mascheroni constant.

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A341749 Numbers k such that gcd(k, phi(k)) > log(log(k)).

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A341750 Numbers k such that gcd(k, sigma(k)) > log(log(k)).

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A125313 Decimal expansion of 2*exp(-gamma).

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A181110 Decimal expansion of 1/zeta(2) - 1/e^gamma, where gamma is the Euler-Mascheroni constant and zeta(2) = Pi^2/6.

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A360895 Decimal expansion of exp(exp(-gamma)) where gamma is the Euler-Mascheroni constant A001620.

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A049012 Composite numbers n such that number of nonprime d with 0 < d < n, gcd(n,d)=1, is pi(n).

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A241532 Decimal expansion of 2e^(-2gamma), gamma being the Euler constant.

A321161 Decimal expansion of Wilf's formula: Product_{k>=1} exp(-1/k)(1 + 1/k + 1/(2k^2)) = exp(-gamma)*cosh(Pi/2)/(Pi/2).