A001692 -id:A001692 - cp's OEIS frontend

A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

1, 3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880, 16104, 44220, 122640, 341484, 956576, 2690010, 7596480, 21522228, 61171656, 174336264, 498111952, 1426403748, 4093181688, 11767874940, 33891544368, 97764009000, 282429535752, 817028131140, 2366564736720, 6863037256208, 19924948267224, 57906879556410

Offset: 0

N. J. A. Sloane

Number of Lyndon words of length n on {1,2,3}. A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts. - John W. Layman, Jan 24 2006
Exponents in an expansion of the Hardy-Littlewood constant Product(1 - (3*p - 1)/(p - 1)^3, p prime >= 5), whose decimal expansion is in A065418: the constant equals Product_{n >= 2} (zeta(n)*(1 - 2^(-n))*(1 - 3^(-n)))^(-a(n)). - Michael Somos, Apr 05 2003
Number of aperiodic necklaces with n beads of 3 colors. - Herbert Kociemba, Nov 25 2016
Number of irreducible harmonic polylogarithms, see page 299 of Gehrmann and Remiddi reference and table 1 of Maître article. - F. Chapoton, Aug 09 2021
For n>=2, a(n) is the number of Hesse loops of length 2*n, see Theorem 22 of Dutta, Halbeisen, Hungerbühler link. - Sayan Dutta, Sep 22 2023
For n>=2, a(n) is the number of orbits of size n of isomorphism classes of elliptic curves under the Hesse derivative, see Theorem 2 of Kettinger link. - Jake Kettinger, Aug 07 2024

			For n = 2 the a(2)=3 polynomials are  x^2+1, x^2+x+2, x^2+2*x+2. - _Robert Israel_, Dec 16 2015

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..2102 (terms 0..200 from T. D. Noe)
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 4th line of Table 1 (p. 6).
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 6.
Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023.
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms. Comput. Phys. Comm. 141 (2001), no. 2, 296-312.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. See Table A.2.
Jake Kettinger, The dynamics of the Hesse derivative on the j-invariant, arXiv:2408.04117 [math.AG], 2024.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Computer Physics Communications, Volume 174, Issue 3, 1 February 2006, Pages 222-240.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag, 1978.
Index entries for sequences related to Lyndon words

Column 3 of A074650.

Cf. A000031, A001037, A001693, A001867, A027375, A027377, A054718, A102660.

with(numtheory): A027376 := n -> `if`(n = 0, 1,
add(mobius(d)*3^(n/d), d = divisors(n))/n):
seq(A027376(n), n = 0..32);

a[0]=1; a[n_] := Module[{ds=Divisors[n], i}, Sum[MoebiusMu[ds[[i]]]3^(n/ds[[i]]), {i, 1, Length[ds]}]/n]
a[0]=1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*3^#&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 01 2015 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,3],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n<1,n==0,sumdiv(n,d,moebius(n/d)*3^d)/n)

a(n) = (1/n)*Sum_{d|n} mu(d)*3^(n/d).
(1 - 3*x) = Product_{n>0} (1 - x^n)^a(n).
G.f.: k = 3, 1 - Sum_{i >= 1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) ~ 3^n / n. - Vaclav Kotesovec, Jul 01 2018
a(n) = 2*A046211(n) + A046209(n). - R. J. Mathar, Oct 21 2021

A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

From Richard R. Forberg, May 22 2023: (Start)
This constant is the asymptotic mean of (phi(n)/n)*(sigma(n)/n), where phi is the Euler totient function (A000010) and sigma is the sum-of-divisors function (A000203).
In contrast, the product of the separate means, mean(phi(n)/n) * mean(sigma(n)/n), converges to 1, with the asymptotic mean(sigma(n)/n) = Pi^2/6 = zeta(2). See A013661.
Also see A062354. (End)

			0.88151383972517077692839182290...

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Steven R. Finch, Class number theory, page 7. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50 and 85.
R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 5, constant Q_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Simon Plouffe, Generalized expansions of real numbers, 2006.
Eric Weisstein's World of Mathematics, Quadratic Class Number Constant.
Eric Weisstein's World of Mathematics, Prime Products.

Cf. A000010, A007434, A013661, A062354, A078087.

$MaxExtraPrecision = 1000; digits = 98; terms = 1000; LR = Join[{0, 0, 0}, LinearRecurrence[{-2, -1, 1, 1}, {-3, 4, -5, 3}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*PrimeZetaP[n-1]/(n-1), {n, 4, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Mar 14 2021

Sum_{n>=1} phi(n)/(n*J(n)) = (this constant)*A013661 with phi()=A000010() and J() = A007434() [Cohen, Corollary 5.1.1]. - R. J. Mathar, Apr 11 2011

A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - Charles R Greathouse IV, Nov 14 2011

The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - Amiram Eldar, Aug 04 2020

			0.2867474284344787341078927127898384...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Juan Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4
Tim Browning, The divisor problem for binary cubic forms, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; arXiv preprint, arXiv:1006.3476 [math.NT], 2010.
Hubert Delange, On some sets of pairs of positive integers, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [cached copy]
László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
László Tóth, Another generalization of Euler's arithmetic function and of Menon's identity, arXiv:2006.12438 [math.NT], 2020. See p. 3.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.

digits = 100; NSum[-(2+(-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 11 2016 *)

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

Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - Michel Marcus, Apr 16 2016
The constant c in Sum_{k<=x} mu(k)^2 * 2^omega(k) = c * x * log(x) + O(x), where mu is A008683 and omega is A001221, and in Sum_{k<=x} 3^omega(k) = (1/2) * c * x * log(x)^2 + O(x*log(x)) (see Tenenbaum, 2015). - Amiram Eldar, May 24 2020
Equals A065472 * A227929 =  A065472 / A098198. - Amiram Eldar, Aug 04 2020

Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003

More digits from Vaclav Kotesovec, Dec 19 2019

A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

Original entry on oeis.org

1, 4, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420

Offset: 0

N. J. A. Sloane

Apart from initial terms, exponents in expansion of A065419 as a product zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 4 colors. - Herbert Kociemba, Nov 25 2016

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..1666 (terms 0..200 from T. D. Noe)
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 3.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
Index entries for sequences related to Lyndon words

Cf. A001037, A027376, A054719, A054660, A054661.

Column k=4 of A074650.

A027377 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s/n); fi; end;

a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}] / n; a[0] = 1; Table[a[n], {n, 0, 23}](* Jean-François Alcover, Nov 29 2011 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,4],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)<<(2*n/d))/n,1) \\ Charles R Greathouse IV, Nov 29 2011

a(n) = Sum_{d|n} mu(d)*4^(n/d)/n.
G.f.: k=4, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) = A054661(n) + 3 * A054660(n). - Andrey Zabolotskiy, Dec 17 2020
a(n) = 2 * (A054664(n) + A054660(n)). - Andrey Zabolotskiy, Dec 19 2020
a(n) = A054719(n)/n, n>0. - R. J. Mathar, Dec 16 2024

A065414 Decimal expansion of rank 2 Artin constant Product_{p prime} (1-1/(p^3-p^2)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.697501358496365903284670350820922924...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products.., arxiv:0903.2514 [math.NT], 2009-2011, variable A_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Index entries for sequences related to Artin's conjecture.

Cf. A005596, A065417.

digits = 99; m0 = 1000; dm = 100; Clear[s]; r[n_] := RootSum[-1 - #^2 + #^3 &, #^n&] - 1; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 300] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], Print[m]; m = m + dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 14 2016 *)

prodeulerrat(1-1/(p^3-p^2)) \\ Amiram Eldar, Mar 12 2021

A065487 Decimal expansion of Product_{p prime} (1 + 1/(p*(p^2-1))).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

			1.2312911488886035627747876512720337...

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A000010, A001615, A003557, A036966, A047994, A072802.

$MaxExtraPrecision = 600; digits = 99; terms = 600; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{0, 2, -1, -1, 1}, {3, 0, 5, -3, 7}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p*(p^2-1))) \\ Amiram Eldar, Mar 17 2021

Equals Sum_{k>=1} 1/A001615(A036966(k)). - Amiram Eldar, Jun 23 2020
Equals Sum_{k>=1} A003557(k)/k^3. - Amiram Eldar, Jan 25 2024
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A047994(k)/A000010(k). - Amiram Eldar, Feb 04 2024

A065484 Decimal expansion of Product_{p prime >= 2} (1 + p/((p-1)^2*(p+1))).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001, Aug 09 2010

Decimal expansion of totient constant. - Eric W. Weisstein, Apr 20 2006

			2.203856596437859787872828316480...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Totient Summatory Function.

Cf. A000010, A002618, A065483, A077387.

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n];
LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]];  (Pi^2/6)*Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]  ] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + p/((p-1)^2*(p+1))) \\ Hugo Pfoertner, Jun 02 2020

Equals Pi^2 * A065483 / 6.
Also defined as: Sum_{m>=1} 1/(m*A000010(m)). See Weisstein link.
Equals 5 * Sum_{m>=1} (-1)^(m+1)/(m*A000010(m)). - Amiram Eldar, Nov 21 2022

A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

Original entry on oeis.org

3, 5, 3, 12, 9, 37, 3, 28, 18, 47, 3, 180, 3, 53, 81, 176, 9, 446, 21, 564, 39, 117, 9, 884, 180, 53, 360, 244, 21, 5959, 9, 800, 39, 111, 369, 9536, 21, 483, 39, 5476, 9, 18289, 9, 1140, 2958, 111, 3, 9424, 6, 3852, 177, 884, 21, 81048, 561, 1188, 69, 227, 9

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). a(n) = number of orders of degree-n monic irreducible polynomials over GF(5).

Also, number of primitive factors of 5^n - 1 (cf. A218357). - Max Alekseyev, May 03 2022

Max Alekseyev, Table of n, a(n) for n = 1..502

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), this sequence (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357
Column k=5 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 13 2024, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ Michel Marcus, Dec 13 2024

a(n) = Sum_{d|n} mu(n/d)*tau(5^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A065483 Decimal expansion of totient constant Product_{p prime} (1 + 1/(p^2*(p-1))).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

The sum of the reciprocals of the cubefull numbers (A036966). - Amiram Eldar, Jun 23 2020

			1.339784153574347246599152586514886052775...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Eric Weisstein's World of Mathematics, Totient Summatory Function

Cf. A036966, A065484, A078074.

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p^2*(p-1))) \\ Vaclav Kotesovec, Sep 19 2020

Equals (6/Pi^2) * A065484. - Amiram Eldar, Jun 23 2020

A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

Original entry on oeis.org

2, -3, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Apart from initial terms, exponents in expansion of A065472 as a product zeta(n)^(-a(n)).

Seiichi Manyama, Table of n, a(n) for n = 1..3000
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
N. J. A. Sloane, Euler transform.

Cf. A001037, A008683, A038064, A038065, A038066, A038067, A038068, A038069, A038070, A059966, A060477, A065472.

a[n_] := DivisorSum[n, (-1)^(#+1) * MoebiusMu[n/#]*2^# &] / n; Array[a, 33] (* Amiram Eldar, May 29 2025 *)

{a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

a(n) = (1/n) * Sum_{d divides n} (-1)^(d+1)*moebius(n/d)*2^d. - Vladeta Jovovic, Sep 06 2002
G.f.: Sum_{n>=1} moebius(n)*log(1 + 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
For n == 0, 1, 3 (mod 4), a(n) = (-1)^(n+1)*A001037(n), which for n>1 also equals (-1)^(n+1)*A059966(n) = (-1)^(n+1)*A060477(n).
For n == 2 (mod 4), a(n) = -(A001037(n) + A001037(n/2)). - George Beck and Max Alekseyev, May 23 2016
a(n) ~ -(-1)^n * 2^n / n. - Vaclav Kotesovec, Jun 12 2018

cp's OEIS Frontend

A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

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A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

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A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

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A065414 Decimal expansion of rank 2 Artin constant Product_{p prime} (1-1/(p^3-p^2)).

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A065487 Decimal expansion of Product_{p prime} (1 + 1/(p*(p^2-1))).

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A065484 Decimal expansion of Product_{p prime >= 2} (1 + p/((p-1)^2*(p+1))).

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