A073004 -id:A073004 - cp's OEIS frontend

A242909 Decimal expansion of exp(-gamma/2).

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

Offset: 0

Jean-François Alcover, May 26 2014

			0.7493060012884490236058715186852615118333...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 3.10 Kneser-Mahler polynomial constants p. 234.

C. J. Smyth, On measures of polynomials in several variables, Bulletin of the Australian Mathematical Society, Volume 23 (1981), Issue 01.

Cf. A073004, A080130, A242908, A242910.

Exp(-EulerGamma(100)/2); // Stefano Spezia, Dec 10 2024

RealDigits[Exp[-EulerGamma/2], 10, 101] // First

Lim_(m->oo) M(z_1, z_2, ..., z_m)/sqrt(m), where M is Mahler's measure for multivariate polynomials.

A244274 Decimal expansion of e*gamma, the product of Euler number and Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 28 2014

			1.569034853003742285079907849123151192307242907588894908656654...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A001113, A001620, A073004, A080130, A244499.

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

RealDigits[E*EulerGamma,10,120][[1]] (* Harvey P. Dale, Jan 01 2015 *)

PARI
```
exp(1)*Euler
```

A246499 Decimal expansion of zeta(2)/exp(gamma), gamma being the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 14 2014

It follows from Mertens theorem that this constant is the limit for large m of log(prime(m))*Product_{k=1..m} 1/(1 + 1/prime(k)).

			0.9235638316741813823235099539877039168469319632611163252035958316...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 5-9

Cf. A001113, A000796, A001620, A013661, A073004, A080130, A097663.

R:=RealField(100); Pi(R)^2/(6*Exp(EulerGamma(R))); // G. C. Greubel, Aug 30 2018

RealDigits[Zeta[2]/E^EulerGamma, 10, 100][[1]] (* Alonso del Arte, Nov 14 2014 *)

PARI
```
Pi^2/6/exp(Euler)
```

Equals Pi^2/(6*exp(gamma)).
Equals lim_{m->infinity} log(prime(m))*Product_{k=1..m} 1/(1 + 1/prime(k)).
Equals A013661/A073004. - Michel Marcus, Nov 18 2014

A061091 Number of k with 1 <= k <= n relatively prime to phi(k).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 28

Offset: 1

Frank Ellermann, May 29 2001

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 2.
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. 12 (1948) 75-78.
Steven R. Finch, Euler Totient Function Asymptotic Constants. [Broken link]
Steven R. Finch, Euler Totient Function Asymptotic Constants. [From the Wayback machine]
Paul Pollack, Numbers which are orders only of cyclic groups, Proceedings of the American Mathematical Society, Vol. 150, No. 2 (2022), pp. 515-524; arXiv preprint, arXiv:2007.09734 [math.NT], 2020.
Imre Z. Ruzsa, Erdős and the integers, J. Number Theory, Vol. 79, No. 1 (1999), 115-163.

Partial sums of A297086.

Cf. A000010 (phi), A001620 (gamma), A003277, A073004, A080130.

s[n_] := Boole[CoprimeQ[n, EulerPhi[n]]]; Accumulate[Array[s, 100]]  (* Amiram Eldar, Dec 10 2024 *)

a(n) = sum(k=1, n, gcd(k, eulerphi(k)) == 1) \\ Charles R Greathouse IV, Jan 29 2013 (corrected by Iain Fox, Dec 25 2017)

list(lim) = {my(s = 0); for(k = 1, lim, s += gcd(k, eulerphi(k)) == 1; print1(s, ", "));} \\ Amiram Eldar, Dec 10 2024

Limit_{n->oo} a(n) * log(log(log(n))) / n = 1/exp(gamma).
a(n) = Sum_{k=1..n} gcd(k, phi(k)) = 1.
a(1) = 1; a(n) = a(n-1) + A297086(n). - Iain Fox, Dec 25 2017

A202412 Decimal expansion of Gamma(gamma).

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Jan 12 2012

Gamma is the Euler Gamma function, gamma is the Euler-Mascheroni constant A001620.

			1.54388174181566169872011938841317019355218...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Euler-Mascheroni constant.
Wikipedia, Gamma function.

Cf. A001620, A073004.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(EulerGamma(R)); // G. C. Greubel, Sep 03 2018

RealDigits[Gamma[EulerGamma], 10, 100][[1]]

default(realprecision, 100); gamma(Euler) \\ G. C. Greubel, Sep 03 2018

gamma*Gamma(gamma) = exp(-gamma^2)*Prod_{n=1..oo} exp(gamma/n)/(1+gamma/n). -- Karl Weierstrass, 1854

A215000 a(n) = floor(exp(1 + 1/2 + 1/3 + ... + 1/n)).

Original entry on oeis.org

2, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 56, 57, 59, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 95, 97, 98, 100, 102, 104, 105, 107, 109, 111, 113, 114

Offset: 1

Clark Kimberling, Aug 18 2012

nonn

a(n) is the greatest integer k for which log k < 1 + 1/2 + ... + 1/n.

a(n) is asymptotically equals to n*e^(gamma) for large values of n where 'gamma' is the Euler-Mascheroni constant (Cf. A001620). - Balarka Sen, Aug 19 2012

			log 2 < 1 < log 3, so a(1) = 2;
log 4 < 1 + 1 + 1/2 < log 5, so a(2) = 4;
log 6 < 1 + 1/2 + 1/3 < log 7, so a(3) = 6.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A215001, A001620, A073004.

[Floor(Exp((&+[1/k :k in [1..n]]))): n in [1..30]]; // G. C. Greubel, Feb 01 2018

f[n_] := Sum[1/h, {h, n}]; Table[Floor[E^f[n]], {n, 100}]
Table[Floor[Exp[HarmonicNumber[n]]], {n, 1, 100}] (* G. C. Greubel, Aug 30 2018 *)

a(n) = floor(exp(sum(X=1,n,1/X))) \\ Balarka Sen, Aug 19 2012

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

Original entry on oeis.org

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

A335004 Decimal expansion of 6*exp(gamma)/Pi^2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 19 2020

			1.0827621932609245801221880381909265701843066555836...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.1, p. 31.
József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, page 100.

J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Jean M. de Koninck and Claude Levesque (eds.), Théorie des nombres/Number theory: Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, Berlin, New York: de Gruyter, 1989, pp. 201-206.
Florian Luca and Carl Pomerance, On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions phi and sigma, Colloquium Mathematicum, Vol. 92 (2002), pp. 111-130.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.

Cf. A001620 (gamma), A013661 (Pi^2/6), A051377 (esigma), A059956 (6/Pi^2), A073004 (exp(gamma)), A246499 (Pi^2/(6*exp(gamma))).

Cf. A236435, A236436.

RealDigits[6*Exp[EulerGamma]/Pi^2, 10, 100][[1]]

6*exp(Euler)/Pi^2 \\ Michel Marcus, May 19 2020

Equals limsup_{k->oo} esigma(k)/(k*log(log(k))), where esigma(k) is the sum of exponential divisors of k (A051377).
Equals A073004 * A059956 = A073004 / A013661 = 1 / A246499.
Equals lim_{k->oo} (1/log(k)) * Product_{p prime <= k} (1 + 1/p). - Amiram Eldar, Jul 09 2020

A342455 The fifth powers of primorials: a(n) = A002110(n)^5.

Original entry on oeis.org

1, 32, 7776, 24300000, 408410100000, 65774855015100000, 24421743243121524300000, 34675383095948798128025100000, 85859681408495723096004822084900000, 552622359415801587878908964592391520700000, 11334919554709059323420895730190266747414284300000, 324509123504618420438174660414872405442002404781629300000

Offset: 0

Antti Karttunen, Mar 12 2021

nonn

The ratio G(n) = sigma(n) / (exp(gamma)*n*log(log(n))), where gamma is the Euler-Mascheroni constant (A001620), as applied to these numbers from a(1)=32 onward, develops as:
0.8893323133
0.7551575418
0.7303870617
0.7347890824
0.7263701246
0.7298051649
0.7304358358
0.7354921494
0.7389343933
0.7391912616
0.7416291350
0.7424159544
...
Notably, after its minimum at term a(5) = 65774855015100000, it starts increasing again, albeit rather slowly. At n=10000 the ratio is 0.8632750..., and at n=40000, it is 0.87545260... Question: Does this trend continue indefinitely? In contrast, for primorials, A002110, the ratio appears to be monotonically decreasing, see comments in A342000.

Young Ju Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), pp. 357-372.
Index entries for sequences related to primorial numbers

Cf. A000584, A001620, A002110, A073004, A181555, A342000.

Diagonal in A079474. After the initial term, also the leftmost branch in that subtree of A329886 whose root is 32.

FoldList[Times, 1, Prime@ Range[11]]^5 (* Michael De Vlieger, Mar 14 2021 *)

PARI
```
A342455(n) = prod(i=1,n,prime(i))^5;
```

from sympy.ntheory.generate import primorial
def A342455(n): return primorial(n)**5 if n >= 1 else 1 # Chai Wah Wu, Mar 13 2021

a(n) = A000584(A002110(n)) = A002110(n)^5.

A346153 a(n) = A346152(n!).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 7, 5, 5, 5, 7, 5, 7, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Amiram Eldar, Jul 07 2021

nonn

Erdős and Selfridge (1982) proved that if f(n) = primepi(a(n)) (or, equivalently, a(n) = prime(f(n))), then |f(n+1) - f(n)| <= 1, and that for infinitely many values of n, f(n+1) = f(n) - 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer., Vol. 34 (1982), pp. 25-45.
Paul Erdős and John L. Selfridge, Problem 6339, Advanced problems, The American Mathematical Monthly, Vol. 88, No. 4 (1981), p. 294; Factorizationf n!, solution to problem 6339, solved by the proposers, ibid., Vol. 89, No. 10 (1982), pp. 790-794.

Cf. A000142, A001620, A073004, A346152.

f[1] = 1; f[n_] := Module[{fct = FactorInteger[n], prods, ind}, prods = Rest @ FoldList[Times, 1, Power @@@ fct]; ind = FirstPosition[prods^2, ?(# > n &)][[1]]; fct[[ind, 1]]]; a[n] := f[n!]; Array[a, 100]

a(n) = A346152(A000142(n)).

Lim_{n->oo} a(n)/sqrt(n) = exp(gamma - 1/2), where gamma is Euler's constant (A001620) (Erdős and Selfridge, 1982).

cp's OEIS Frontend

A242909 Decimal expansion of exp(-gamma/2).

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A244274 Decimal expansion of e*gamma, the product of Euler number and Euler-Mascheroni constant.

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A246499 Decimal expansion of zeta(2)/exp(gamma), gamma being the Euler-Mascheroni constant.

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A061091 Number of k with 1 <= k <= n relatively prime to phi(k).

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A202412 Decimal expansion of Gamma(gamma).

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A215000 a(n) = floor(exp(1 + 1/2 + 1/3 + ... + 1/n)).

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

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