A080130 -id:A080130 - cp's OEIS frontend

A073004 Decimal expansion of exp(gamma).

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

See references and additional links in A094644.
The Riemann hypothesis holds if and only if the inequality sigma(n)/(n*log(log(n))) < exp(gamma) is valid for all n >= 5041, (G. Robin, 1984). - Peter Luschny, Oct 18 2020
From Peter Bala, Aug 24 2025: (Start)
By definition, gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*exp(s(n+k)). Then it appears that E(n) converges rapidly to exp(gamma). For example, E(50) = 1.78107241799019798523650410310(43...) gives exp(gamma) correct to 29 decimal digits. Cf. A002389. (End)

			Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.5.1 and 2.27.2, pp. 31, 187.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 166, 191, 208.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
C. Elsner, On a sequence transformation with integral coefficients for Euler's constant, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
Paul Erdős and S. K. Zaremba, The arithmetic function Sum_{d|n} log d/d, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579.
T. H. Gronwall, Some Asymptotic Expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.
Simon Plouffe, The exp(gamma).
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Gronwall's Theorem.
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 2-3.
Eric Weisstein's World of Mathematics, Robin's Theorem.

Cf. A001620 (Euler-Mascheroni constant, gamma).

Cf. A001113, A002389, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A155969, A246499.

R:=RealField(100); Exp(EulerGamma(R)); // G. C. Greubel, Aug 27 2018

RealDigits[ E^(EulerGamma), 10, 110] [[1]]

PARI
```
exp(Euler)
```

By Mertens theorem, equals lim_{m->infinity}(1/log(prime(m))*Product_{k=1..m} 1/(1-1/prime(k))). - Stanislav Sykora, Nov 14 2014
Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - Amiram Eldar, Nov 07 2020
Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - Amiram Eldar, Mar 03 2021
Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - Amiram Eldar, Mar 20 2022
Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - Thomas Ordowski, Jan 30 2023
Equals Product_{k>=1} (k/sqrt(2))^((-1)^k/(k*log(2))). - Antonio Graciá Llorente, Oct 11 2024
Equals lim_{n->oo} (1/log(n))*Product_{prime p<=n} p/(p - 1) [Mertens] (see Finch at p. 31). - Stefano Spezia, Oct 27 2024

A007838 Number of permutations of n elements with distinct cycle lengths.

Original entry on oeis.org

1, 1, 1, 5, 14, 74, 474, 3114, 24240, 219456, 2231280, 23753520, 288099360, 3692907360, 51677246880, 775999798560, 12364465397760, 208583679951360, 3770392002048000, 71251563061002240, 1421847102467635200, 29861872557056870400, 655829140087057305600

Offset: 0

Arnold Knopfmacher

nonn

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., Birkhäuser, Boston, 1982.

Seiichi Manyama, Table of n, a(n) for n = 0..450 (terms 0..200 from Vincenzo Librandi)
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006.
A. Knopfmacher and R. Warlimont, Counting permutations and polynomials with a restricted factorization pattern, Australasian J. of Combinatorics, 13 (1996), 151-162.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Examples 8.10 and 11.8 (pdf, ps)

Cf. A000142, A080130, A087639, A088994, A317166.

p := product((1+x^m/m), m=1..100): s := series(p,x,100): for i from 1 to 100 do printf(`%.0f,`,i!*coeff(s,x,i)) od:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 23 2022

max = 20; p = Product[(1 + x^m/m), {m, 1, max}]; s = Series[p, {x, 0, max}]; CoefficientList[s, x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011, after Maple *)

{a(n)=if(n<0, 0, n!*polcoeff( prod(k=1, n, 1+x^k/k, 1+x*O(x^n)), n))} /* Michael Somos, Sep 19 2006 */

E.g.f.: Product_{m >= 1} (1+x^m/m).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)^(1-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
Asymptotics: a(n) ~ n!(e^{-g} + e^{-g}/n + O((log n)/n^2)), where g is the Euler gamma.
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018

More terms from James Sellers, Dec 24 1999

A004061 Numbers k such that (5^k - 1)/4 is prime.

Original entry on oeis.org

3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593

Offset: 1

N. J. A. Sloane

With the addition of the 19th prime in the sequence, the new best linear fit to the sequence has G=0.4723, which is slightly closer to the conjectured limit of G=0.56145948, A080130 (see link for Generalized Repunit Conjecture). - Paul Bourdelais, Apr 30 2018

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paul Bourdelais, A Generalized Repunit Conjecture [From _Paul Bourdelais_, Jun 01 2010]
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n-1)/k

Cf. A080130.

lst={};Do[If[PrimeQ[(5^n-1)/4], AppendTo[lst, n]], {n, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 20 2008 *)

forprime(p=2,1e4,if(ispseudoprime(5^p\4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(13)-a(15) from Kamil Duszenko (kdusz(AT)wp.pl), Mar 25 2003
a(16) corresponds to a probable prime based on trial factoring to 4*10^13 and Fermat primality testing base 2. - Paul Bourdelais, Dec 11 2008
a(17) corresponds to a probable prime discovered by Paul Bourdelais, Jun 01 2010
a(18) corresponds to a probable prime discovered by Paul Bourdelais, Apr 30 2018
a(19) corresponds to a probable prime discovered by Ryan Propper, Jan 02 2022

A048855 Number of integers up to n! relatively prime to n!.

Original entry on oeis.org

1, 1, 1, 2, 8, 32, 192, 1152, 9216, 82944, 829440, 8294400, 99532800, 1194393600, 16721510400, 250822656000, 4013162496000, 64210599936000, 1155790798848000, 20804234379264000, 416084687585280000, 8737778439290880000, 192231125664399360000

Offset: 0

Paul Max Payton

Rephrasing the Quet formula: Begin with 1. Then, if n + 1 is prime subtract 1 and multiply. If n+1 is not prime, multiply. Continue writing each product. Thus the sequence would begin 1, 2, 8, . . . . The first product is 1*(2 - 1), second is 1*(3 - 1), and third is 2*4. - Enoch Haga, May 06 2009

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, page 134.

Charles R Greathouse IV, Table of n, a(n) for n = 0..450
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A000010, A000142, A014197.

Cf. A001620, A080130.

with(numtheory):a:=n->phi(n!): seq(a(n), n=0..20); # Zerinvary Lajos, Oct 07 2007

Table[ EulerPhi[ n! ], {n, 0, 21}] (* Robert G. Wilson v, Nov 21 2003 *)

a(n)=eulerphi(n!) \\ Charles R Greathouse IV, May 12 2011

from math import factorial, prod
from sympy import primerange
from fractions import Fraction
def A048855(n): return (factorial(n)*prod(Fraction(p-1,p) for p in primerange(n+1))).numerator # Chai Wah Wu, Jul 06 2022

[euler_phi(factorial(n)) for n in range(0,21)] # Zerinvary Lajos, Jun 06 2009

a(n) = phi(n!) = A000010(n!).
If n is composite, then a(n) = a(n-1)*n. If n is prime, then a(n) = a(n-1)*(n-1). - Leroy Quet, May 24 2007
Under the Riemann Hypothesis, a(n) = n! / (e^gamma * log n) * (1 + O(log n/sqrt(n))). - Charles R Greathouse IV, May 12 2011
Sum_{k=1..n} a(k) = exp(-gamma) * (n!/log(n)) * (1 + O(1/log(n)^3)), where gamma is Euler's constant (A001620) (De Koninck and Verreault, 2024, p. 56, eq. (4.12)). - Amiram Eldar, Dec 10 2024

Name changed by Daniel Forgues, Aug 01 2011

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.

A322364 Numerator of the sum of inverse products of parts in all partitions of n.

Original entry on oeis.org

1, 1, 3, 11, 7, 27, 581, 4583, 2327, 69761, 775643, 147941, 30601201, 30679433, 10928023, 6516099439, 445868889691, 298288331489, 7327135996801, 1029216937671847, 14361631943741, 837902013393451, 2766939485246012129, 274082602410356881, 835547516381094139939

Offset: 0

Alois P. Heinz, Dec 04 2018

			1/1, 1/1, 3/2, 11/6, 7/3, 27/10, 581/180, 4583/1260, 2327/560, 69761/15120, 775643/151200, 147941/26400, 30601201/4989600, 30679433/4633200 ... = A322364/A322365

Alois P. Heinz, Table of n, a(n) for n = 0..505
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.

Denominators: A322365.

Cf. A000041, A006906, A080130, A177208, A177209, A322380, A322381, A323290, A323291, A323339, A323340.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +b(n-i, min(i, n-i))/i)
    end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n==0||i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]/i];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

a(n) = {my(s=0); forpart(p=n, s += 1/vecprod(Vec(p))); numerator(s);} \\ Michel Marcus, Apr 29 2020

Limit_{n-> infinity} a(n)/(n*A322365(n)) = exp(-gamma) = A080130.

A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

Original entry on oeis.org

1, 1, 1, 5, 7, 37, 79, 173, 101, 127, 1033, 1571, 200069, 2564519, 5126711, 25661369, 532393, 431100529, 1855391, 1533985991, 48977868113, 342880481117, 342289639579, 435979161889, 1308720597671, 373092965489, 7824703695283, 24141028973, 31250466692609

Offset: 0

Alois P. Heinz, Dec 05 2018

a(n)/A322381(n) = A007838(n)/A000142(n) is the probability that a random permutation of [n] has distinct cycle sizes. - Geoffrey Critzer, Feb 23 2022

			1/1, 1/1, 1/2, 5/6, 7/12, 37/60, 79/120, 173/280, 101/168, 127/210, 1033/1680, 1571/2640, 200069/332640, 2564519/4324320, 5126711/8648640, ... = A322380/A322381

Alois P. Heinz, Table of n, a(n) for n = 0..1268
Andreas B. G. Blobel, An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k), arXiv:1904.07808 [math.CO], 2019.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 137.
A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.

Denominators: A322381.

Cf. A000009, A000142, A007838, A022629, A080130, A177208, A177209, A322364, A322365, A323290, A323291, A323339, A323340.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

Limit_{n->infinity} a(n)/A322381(n) = exp(-gamma) = A080130.

Sum_{n>=0} a(n)/A322381(n)*x^n = Product_{i>=1} (1 + x^i/i). - Geoffrey Critzer, Feb 23 2022

A079650 Continued fraction for e^(-gamma).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 3, 5, 4, 1, 1, 2, 2, 1, 7, 9, 1, 16, 1, 1, 1, 2, 6, 1, 2, 1, 6, 2, 59, 1, 1, 1, 3, 3, 3, 2, 1, 3, 5, 100, 1, 58, 1, 2, 1, 94, 1, 1, 2, 2, 10, 1, 2, 7, 1, 3, 4, 5, 3, 10, 1, 21, 1, 11, 1, 4, 1, 2, 2, 1, 2, 2, 1, 8, 3, 2, 1, 1, 6, 1, 2, 2, 1, 38, 2, 1, 4, 1, 3, 1, 1, 5, 3, 1, 52, 1, 2, 2

Offset: 0

Neil Fernandez, Jan 22 2003

Same as A094644 except this number begins with a 0. - T. D. Noe, Jun 18 2012

			e^(-gamma) = 0.561... = 0 + 1/(1+ 1/ (1 +1/(3+...))), so sequence begins 0, 1, 1, 3,...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A080130 (exp(-gamma)), A094644.

ContinuedFraction[Exp[-EulerGamma], 100] (* Paolo Xausa, Aug 07 2024 *)

contfrac(exp(-Euler)) \\ Michel Marcus, Oct 13 2019

Offset changed by Andrew Howroyd, Aug 07 2024

A096987 Numerator of Sum_{k=1..n} 1/H(k), where H(k) = Sum_{j=1..k} 1/j is the k-th harmonic number.

Original entry on oeis.org

0, 1, 5, 73, 2221, 353777, 19595573, 239046803, 198972350083, 1535302297058707, 100536661265514127, 8974880059175708288297, 818810519369821323965929237, 990666575600755815615137883006341, 1220749860499992165560973207703210595953

Offset: 0

Leroy Quet, Aug 19 2004

			1/1 + 1/(1 + 1/2) + 1/(1 + 1/2 + 1/3) = 73/33, so a(3) = 73.

Cf. A124432 (denominators), A000720, A001008, A002805, A001620, A080130.

f[n_] := Numerator[ Sum[ 1/HarmonicNumber[j], {j, 1, n}]]; Table[ f[n], {n, 0, 14}] (* Robert G. Wilson v, Aug 21 2004 *)

m=13;for(n=0,m,print1(numerator(sum(k=1,n,1/sum(j=1,k,1/j))),",")) \\ Klaus Brockhaus, Aug 21 2004

From Thomas Ordowski, Mar 21 2023: (Start)
Sum_{k=1..n} 1/H(k) ~ Sum_{k=2..n} 1/log(k) ~ Integral_{2..n} dx/log(x) = Li(n).
Sum_{k=1..n} 1/H(k) = Sum_{k=1..n} 1/(log(k + 1/2) + gamma) - C + o(1), where gamma = A001620 = 0.577... is Euler's constant and the constant C = 0.0229825...
Sum_{k=1..n} 1/H(k) = exp(-gamma)*(Ei(log(n) + gamma) - 1) + o(1), where Ei(x) is the exponential integral function of real x, and we have Ei(log(x)) = li(x).
Note that a(n)/A124432(n) ~ pi(n) = A000720(n), see my first formula.
Sum_{k=1..n} 1/H(k) = n/(H(n) - 1 + ...) = n/(log(n) + gamma - 1 + O(1/log(n))).
Theorem: lim_{n->oo} (H(n) - n / Sum_{k=1..n} 1/H(k)) = 1, see my third formula.
Proof: since Integral dx / (log(x) + gamma) = exp(-gamma)*Ei(log(x) + gamma) + c, so we get lim_{n->oo} (log(n) + gamma - n*exp(gamma) / Ei(log(n) + gamma)) = 1, qed. (End)

More terms from Klaus Brockhaus and Robert G. Wilson v, Aug 21 2004

A227242 Decimal expansion of (e^gamma - 1)/e^gamma.

Original entry on oeis.org

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

Offset: 0

Arkadiusz Wesolowski, Oct 19 2013

The value is equal to lim_{n->oo} (Sum_{d|n#, d>n} 1/phi(d))/(Sum_{d|n#} 1/phi(d)).

			(exp(gamma) - 1)/exp(gamma) = 0.438540516433114830175856785....

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000142, A001113, A001620, A073004, A080130, A351901.

E:=EulerGamma(RealField(105)); Reverse(Intseq(Floor(10^105*(Exp(E)-1)/Exp(E))));

evalf(1-exp(-gamma), 120);  # Alois P. Heinz, Feb 24 2022

RealDigits[(E^EulerGamma - 1)/E^EulerGamma, 10, 50][[1]] (* G. C. Greubel, Oct 02 2017 *)

default(realprecision, 105); x=10*(exp(Euler)-1)/exp(Euler); for(n=1, 105, d=floor(x); x=(x-d)*10; print1(d, ", "));

From Alois P. Heinz, Feb 24 2022: (Start)
Equals 1 - exp(-gamma) = 1 - A080130.
Equals lim_{n->oo} A351901(n)/A000142(n). (End)

cp's OEIS Frontend

A073004 Decimal expansion of exp(gamma).

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A007838 Number of permutations of n elements with distinct cycle lengths.

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A004061 Numbers k such that (5^k - 1)/4 is prime.

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A048855 Number of integers up to n! relatively prime to n!.

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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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Mathematica

Formula

A322364 Numerator of the sum of inverse products of parts in all partitions of n.

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Maple

Mathematica

PARI

Formula

A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

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