Juan Arias-de-Reyna - cp's OEIS frontend

A288163 Integers related to the half moments of Rvachëv function.

1, 1, 5, 84, 4004, 494760, 150120600, 107969547840, 179605731622464, 678695382464158080, 5745964983105758544000, 107798142804281290451059200, 4441362930723337358985334172160, 398854836980938754158182857661404160, 77576833096847783279235708819073596288000

Offset: 0

Juan Arias-de-Reyna, Jun 06 2017

nonn

These numbers determine the half moments of the Rvachëv function. The Rvachëv function is related to the Fabius function, up(x)=F(x+1) for |x|<1 and up(x)=0 for |x|>=1.

J. Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A287936, A287937, A287938, A288161.

d[0] = 1;
d[n_] := d[n] =
  Sum[Binomial[n + 1, k] d[k], {k, 0, n - 1}]/((n + 1)*(2^n - 1));
a[n_] := (n + 1)! Product[(2^k - 1), {k, 1, n}] d[n];
Table[a[n], {n, 0, 14}]

a(n) = (n+1)!*Product_{k=1..n}(2^k-1)*d(n) where d(n) are the rationals defined by the recurrence d(0)=1; d(n)=Sum_{k=0..n-1}[binomial(n+1,k)d(k)]/((n+1)*(2^n-1)) (cf. A288161).

A287938 Integers associated with moments of Rvachëv function.

Original entry on oeis.org

1, 1, 19, 2915, 2788989, 14754820185, 402830065455939, 54259734183964303995, 34931036957548128175343565, 104968042559556881090071537121985, 1445701512369903326110289606343988638195, 89942525814858602265845303890518923811304544595, 24979493321562411847493262443987087581059026281953954525

Offset: 0

Juan Arias-de-Reyna, Jun 03 2017

nonn

a(n) is equal to the product of (2*n+1)!!*(Product_{k=1..n} (2^(2*k)-1)) and A287936(n)/A287937(n), the moment of the Rvachëv function. The Rvachëv function is related to the Fabius function; up(x)=F(x+1) for |x|<1 and up(x)=0 for |x|>=1, where F is the Fabius function.

J. Arias de Reyna, An infinitely differentiable function with compact support:Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A272755, A272757, A287936, A287937.

c[0] = 1;
c[n_] := c[n] =
   Sum[Binomial[2 n + 1, 2 k] c[k], {k, 0, n - 1}]/((2 n + 1) (2^(2 n) - 1));
a[n_] := a[n] = c[n] (2 n + 1)!! Product[(2^(2 k) - 1), {k, 1, n}];
Table[a[n], {n, 0, 30}]
Table[(-1)^n 4^(-n) (2 n)! (2 n + 1)!! Sum[QBinomial[n, k, 1/4] 2^(-k (3 k + 1)/2)/(2 n + k + 1)! Sum[(-1)^ThueMorse[m] (2 m + 1)^(2 n + k + 1), {m, 0, 2^k - 1}], {k, 0, n}], {n, 0, 12}] (* Vladimir Reshetnikov, Jul 08 2018 *)

a(n) = (2*n+1)!!*(Product_{k=1..n} (2^(2*k)-1))*A287936(n)/A287937(n).

A287936 Numerator of moments of Rvachëv function up(x).

Original entry on oeis.org

1, 1, 19, 583, 132809, 46840699, 4068990560161, 1204567303451311, 4146897304424408411, 18814360006695807527868793, 21431473463327429953796293981397, 911368783375270623395381542054690099, 3805483535214088799368825731508632105336401423

Offset: 0

Juan Arias-de-Reyna, Jun 03 2017

a(n)/A287937(n) is equal to the integral of t^(2n) * up(t), the moment of the Rvachëv function. The Rvachëv function is related to the Fabius function; up(x)=F(x+1) for |x|<1 and up(x)=0 for |x|>=1, where F is the Fabius function.

J. Arias de Reyna, An infinitely differentiable function with compact support:Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A287937, A287938.

c[0] = 1;
c[n_] := c[n] =
   Sum[Binomial[2 n + 1, 2 k] c[k], {k, 0, n - 1}]/((2 n + 1) (2^(2 n) - 1));
Table[Numerator[c[n]], {n, 0, 30}]

Recurrence c(0)=1, c(n)=Sum_{k=0..n-1}(binomial(2n+1,2k) c_k)/((2n+1)*(2^(2n)-1)), where c(n)=a(n)/A287937(n).

A288161 Denominator of half moments of Rvachëv function.

Original entry on oeis.org

2, 18, 6, 1350, 270, 23814, 17010, 65063250, 7229250, 9762090030, 4437313650, 8267713725521250, 635977978886250, 81188783595533250, 297692206516955250, 22510683177794610356250, 1564913803803903393750, 40011216302189267004656036250, 10529267447944543948593693750

Offset: 1

Juan Arias-de-Reyna, Jun 06 2017

a(n) is equal to the denominator of the integral over (0,1) of n*t^(n-1)*up(t).
These numbers are the half moments of the Rvachëv function. The Rvachëv function is related to the Fabius function, up(x)=F(x+1) for |x|<1 and  up(x)=0 for |x|>=1.
The sequence of numerators is not in the OEIS because it appears t coincide with A272755: Numerators of Fabius function F(1/2^n). In fact d(n) = n! 2^binomial(n,2)F(1/2^n). The coincidence depends on the fact that n! 2^binomial(n,2) divides the denominator of F(1/2^n). It is true that 2^binomial(n,2) divides this denominator, but I do not see any reason for n! to divide this denominator.

			The rationals d(n) are  1/2, 5/18, 1/6, 143/1350, 19/270,  ...

J. Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A287936, A287937, A287938.

d[0] = 1;
d[n_] := d[n] =
  Sum[Binomial[n + 1, k] d[k], {k, 0, n - 1}]/((n + 1)*(2^n - 1));
Table[Denominator[d[n]], {n, 1, 20}]

Recurrence d(0)=1; d(n)=Sum_{k=0..n-1}(binomial(n+1,k)d(k))/((n+1)*(2^n-1)) with a(n) are the denominators of d(n).

It may also be defined to be the only sequence d(n) with d(0)=1 and such that the function f(x)=Sum_{n>=0} d(n) x^n/n! satisfies x*f(2x)=(e^x-1)*f(x).

A287937 Denominator of moments of Rvachëv function up(x).

Original entry on oeis.org

1, 9, 675, 59535, 32531625, 24405225075, 4133856862760625, 2232691548877164375, 13301767332333178846875, 100028040755473167511640090625, 182171989134769427819794434994453125, 12012265189685856975048179723754213046875, 75749878923357625026812035792140968086378130859375

Offset: 0

Juan Arias-de-Reyna, Jun 03 2017

			A287936(n)/a(n) = 1/1, 1/9, 19/675, 583/59535, ...

J. Arias de Reyna, An infinitely differentiable function with compact support:Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A287936, A287938.

c[0] = 1;
c[n_] := c[n] =
   Sum[Binomial[2 n + 1, 2 k] c[k], {k, 0, n - 1}]/((2 n + 1) (2^(2 n) - 1));
Table[Denominator[c[n]], {n, 0, 30}]

Recurrence c(0)=1, c(n)=Sum_{k=0..n-1}(binomial(2n+1,2k) c_k)/((2n+1)*(2^(2n)-1)), where c(n)=A287936(n)/a(n).

A281391 Vinogradov's number J_{3,2}(n).

Original entry on oeis.org

1, 20, 93, 256, 563, 1032, 1771, 2744, 4077, 5788, 7985, 10560, 13855, 17600, 22047, 27304, 33425, 40140, 47989, 56504, 66315, 77296, 89411, 102336, 117061, 132956, 150201, 168904, 189479, 211080, 235111, 260240, 287385, 316420, 347237

Offset: 1

Juan Arias-de-Reyna, Jan 21 2017

nonn

a(n) is the number of solutions of the system of equations x_1 + x_2 + x_3 = y_1 + y_2 + y_3, x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2 and such that 1 <= x_1, x_2, x_3, y_1, y_2, y_3 <= n.
Vinogradov's numbers J_{s,k}(X) play an important role in many number-theoretic problems, for example Waring's problem and bounds on the zeta function.
There is an asymptotic formula: the first term is due to Rogovskaya, the second to Blomer and Brüdern.

			The system of equations have trivial solutions in which {y_1,y_2,y_3} is a permutation of {x_1,x_2,x_3}. The first nontrivial solutions are in the case J_{3,2}(5), where there are 18 solutions from permutations of {x_1,x_2,x_3} = {1,4,4}, {y_1,y_2,y_3} = {2,2,5}.

Rogovskaya, N. N., An asymptotic formula for the number of solutions of a system of equations, Diophantine Approximations, Part II, Moskov, Gos. Univ., Moscow, 1986, pp. 78-84.

Giovanni Resta, Table of n, a(n) for n = 1..500
V. Blomer and J. Brüdern, The number of integer points on Vinogradov's quadric, Monatsh. Math. 160 (2010) 91-107.
Trevor D. Wooley, Translation invariance, exponential sums, and Waring's problem, arXiv:1404.3508 [math.NT], 2014.

J32[X_] := Module[{T, n, count, P, S, PS, long, K, L, m},
   T = Table[n, {n, 1, X}];
   count = 0;
   P = Tuples[T, 3];
   For[S = 3, S <= 3 X, S++,
    PS[S] = Select[P, Total[#] == S &]];
   For[S = 3, S <= 3 X, S++,
    long = Length[PS[S]];
    For[n = 1, n <= long, n++,
     K = PS[S][[n]];
     For[m = 1, m <= long, m++,
      L = PS[S][[m]];
      If[Total[K^2] == Total[L^2], count = count + 1]]];
    ];
   count];
Table[J32[n], {n, 1, 12}]
(* or *)
a[n_] := Sum[Block[{p,w,e}, p = IntegerPartitions[s, {3}, Range@ n]; w = Length /@ Permutations /@ p; e = (Plus @@ Last /@ #) & /@ GatherBy[ Transpose@ {Plus @@@ (p^2), w}, First]; Total[e^2]], {s, 3, 3 n}]; Array[a, 50] (* faster, Giovanni Resta, Mar 12 2017 *)

a(n) ~ (18/Pi^2)(n^3*log n) + (3/Pi^2)*(12*C - 6zeta'(2)/zeta(2) - 5)*n^3 + O(n^(5/2)log n), where C is Euler's constant.

A271727 Decimal expansion of the density of exponentially 2^n-numbers (A138302).

Original entry on oeis.org

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

Offset: 0

Juan Arias-de-Reyna, Apr 13 2016

			0.87249717935391281355800771433253186691958393977733373765412...

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Density of A138302.

Cf. A271726 (Expansion of log(f(x))).

$MaxExtraPrecision = m = 500; em = 10; f[x_] := Log[1 - x^3 + Sum[x^(2^e) - x^(1 + 2^e), {e, 2, em}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]] (* Amiram Eldar, Sep 09 2022 *)

Equals Product_{prime p} f(1/p), where f(x) = 1-x^3+Sum_{n>=2}(x^(2^n)-x^(1+2^n)).

A271726 Let f(x) = 1 -x^3+ Sum_{j>=2} (x^(2^j)-x^(1+2^j)). Then a(n) is n times the coefficient of x^n in the expansion of log(f(x)).

Original entry on oeis.org

0, 0, -3, 4, -5, -3, 7, -4, -3, 5, -11, 1, 13, -21, 7, 28, -51, 33, 19, -91, 109, -33, -115, 209, -155, -65, 321, -381, 87, 407, -713, 476, 349, -1207, 1227, -35, -1739, 2603, -1277, -1979, 4797, -4161, -903, 7451, -9713, 3427, 9165, -18575, 14021, 6455, -29991, 34779

Offset: 1

Juan Arias-de-Reyna, Apr 13 2016

sign

Function f(x) is connected with the density h of exponentially (2^n)-numbers (A138302). Specifically, for h = Product_{prime p} f(1/p), this sequence allows the calculation of h with very high accuracy (cf. A271727).

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..5000
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

M = 6; K = 50; (* To get the first  50 terms *)
f = 1 - x^3 + Sum[x^(2^r) - x^(2^r + 1), {r, 2, M}];
S = Series[Log[f], {x, 0, K}];
If[2^M <= K, Print["Warning: 2^M should be greater than K and it is not. Change parameters."]];
L = CoefficientList[S, x];
A271726[n_] := n L[[n + 1]];
Table[A271726[n], {n, 1, K}]

A262400 Let f(x) = 1 + Sum_{j>=4} (|mu(j)| - |mu(j-1)|)*x^j, where mu(n) is the Möbius function (A008683). Then a(n) is n times the coefficient of x^n in the expansion of log(f(x)).

Original entry on oeis.org

0, 0, 0, 0, -4, 5, 0, 0, -12, 9, 5, 0, -28, 39, 0, -10, -60, 102, -45, 0, -119, 252, -132, 0, -252, 580, -403, 9, -424, 1363, -1210, 248, -828, 3003, -3332, 1195, -1729, 6697, -8740, 4290, -3807, 14514, -22176, 13889, -9288, 31049, -54142, 41501, -25260, 66885, -129570

Offset: 0

Juan Arias-de-Reyna, Sep 21 2015

sign

Function f(x) is connected with the density h of the exponentially squarefree numbers (A209061). Specifically, for h = Product_{prime p} f(1/p), this sequence allows the calculation of h with very high accuracy (cf. A262276).

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..3000

Cf. A008683, A209061, A262276.

M = 50; (* to get the first 51 terms *)
f = 1 + Sum[(MoebiusMu[n]^2 - MoebiusMu[n - 1]^2) x^n, {n, 4, M}];
S = Series[Log[f], {x, 0, M}];
A262400[nn_] := CoefficientList[S, x][[nn + 1]] nn;
Table[A262400[n], {n, 0, M}]

A262276 Decimal expansion of Toth's constant (the density of the exponentially squarefree numbers).

Original entry on oeis.org

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

Offset: 0

Juan Arias-de-Reyna, Sep 19 2015

			0.95592301586190237688406538670987007467715943165456868832805...

Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.
László Tóth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166.

Density of A209061.

Cf. A008683.

$MaxExtraPrecision = m = 1000; f[x_] := Log[1 - x^4 + (1 - x)*Sum[x^e*(MoebiusMu[e]^2), {e, 4, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 163][[1]] (* Amiram Eldar, Apr 27 2025 *)

Equals Product_{prime p} (1+Sum_{j>=4} (mu(j)^2 - mu(j-1)^2)/p^j), where mu(n) is the Möbius function.

cp's OEIS Frontend

User: Juan Arias-de-Reyna

A288163 Integers related to the half moments of Rvachëv function.

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A287938 Integers associated with moments of Rvachëv function.

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A287936 Numerator of moments of Rvachëv function up(x).

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A288161 Denominator of half moments of Rvachëv function.

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A287937 Denominator of moments of Rvachëv function up(x).

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A281391 Vinogradov's number J_{3,2}(n).

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A271727 Decimal expansion of the density of exponentially 2^n-numbers (A138302).

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A271726 Let f(x) = 1 -x^3+ Sum_{j>=2} (x^(2^j)-x^(1+2^j)). Then a(n) is n times the coefficient of x^n in the expansion of log(f(x)).

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