A244009 -id:A244009 - cp's OEIS frontend

A377588 Decimal expansion of 7zeta(3)/(2Pi^2) - log(2) + 1/2.

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.233131218257560481506289305137990304982506635269...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A094642, A164102, A244009, A377589, A377592.

RealDigits[7Zeta[3]/(2Pi^2)-Log[2]+1/2,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((k + 1)*2^(2*k)) (see Finch).

A377589 Decimal expansion of 9zeta(3)/(2Pi^2) - log(2) + 1/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.18825837982446689796062876035594274490384190278...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A094642, A164102, A244009, A377588, A377592.

RealDigits[9Zeta[3]/(2Pi^2)-Log[2]+1/3,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((2*k + 3)*2^(2*k-1)) (see Finch).

A377592 Decimal expansion of 9zeta(3)/Pi^2 - 93zeta(5)/(2*Pi^4) - log(2) + 1/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.158000963625557733268629385978458549091780284796...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A002388, A013663, A092425, A094642, A244009, A377588, A377589.

RealDigits[9Zeta[3]/Pi^2-93Zeta[5]/(2Pi^4)-Log[2]+1/4,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((k + 2)*2^(2*k)) (see Finch).

A263354 Decimal expansion of the generalized hypergeometric function 3F2(1/2,3/2,3/2; 5/2,5/2;x) at x=1/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 16 2015

			1.113463780929017366397176...

R. J. Mathar, Yet another table of integrals, arXiv:1207.5845 [math.CA], 2012-2024, Eq. (9.82).

Cf. A000796, A010487, A244009, A247685.

evalf(hypergeom([1/2,3/2,3/2],[5/2,5/2],1/2)) ;

RealDigits[9*(4*Catalan - 2 + Pi*(Log[2] - 1))/(4*Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Aug 23 2024 *)

Equals 9*(4*Catalan-2+Pi*(log 2 -1))/(4*sqrt(2)) = 9*(A247685 -2 - A000796 * A244009) / A010487.

A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

Original entry on oeis.org

2, 1, 0, 3, 6, 5, 4, 7, 10, 9, 8, 11, 14, 13, 12, 15, 18, 17, 16, 19, 22, 21, 20, 23, 26, 25, 24, 27, 30, 29, 28, 31, 34, 33, 32, 35, 38, 37, 36, 39, 42, 41, 40, 43, 46, 45, 44, 47, 50, 49, 48, 51, 54, 53, 52, 55, 58, 57, 56, 59, 62, 61, 60, 63, 66, 65, 64

Offset: 0

Wesley Ivan Hurt, Oct 17 2015

Fixed points are the odd numbers (A005408).

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A004442, A005408, A092486, A132429, A176742, A244009, A263449.

[n+(1+(-1)^n)*(-1)^(n*(n+1) div 2) : n in [0..80]];

/* By definition: */ &cat[[4*k+2,4*k+1,4*k,4*k+3]: k in [0..20]]; // Bruno Berselli, Nov 08 2015

A263426:=n->n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2): seq(A263426(n), n=0..80);

Table[n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2), {n, 0, 80}]

Vec((2-3*x+2*x^2+x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015

G.f.: (2 - 3*x + 2*x^2 + x^3)/((x - 1)^2*(1 + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.
a(n) = n + (1 + (-1)^n)*(-1)^(n*(n+1)/2).
a(n) = 4*floor((n+1)/4) - (n mod 4) + 2.
a(n) = A092486(n) - 1.
a(n) = n + A176742(n) for n>0.
a(2n) = 2*A004442(n), a(2n+1) = A005408(n).
a(-n-1) = -A263449(n).
a(n+1) = a(n) - A132429(n+1)*(-1)^n.
Sum_{n>=0, n!=2} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Dec 25 2023

A309638 Nearest integer to 1/F(1/x), where F(x) is the Dickman function.

Original entry on oeis.org

1, 3, 21, 204, 2819, 50891, 1143423, 30939931, 984011503, 36098843631, 1504934136432, 70436763188525, 3664092112471681, 210056231435360023, 13175390260774094846, 898537704166507324228, 66265550246147429710863, 5259409287834480235626661, 447341910388133084658686126, 40620967386538406952534036284

Offset: 1

Jeremy Tan, Aug 11 2019

nonn

The asymptotic density of the n-th-root-smooth numbers is approximately 1/a(n).

Van de Lune and Wattel show a(n) >= A001147(n) for n >= 1.

			The asymptotic density of fifth-root-smooth numbers is F(1/5) = 0.000354724700... = 1/2819.08758..., so a(5) = 2819.

G. Marsaglia, A. Zaman and J. Marsaglia (1989), Numerical Solution of Some Classical Differential-Difference Equations, Mathematics of Computation, 53 (187), 191-201.
Jeremy Tan, Python program
J. van de Lune and E. Wattel (1969), On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory, Mathematics of Computation, 23 (106), 417-421.
Eric Weisstein's World of Mathematics, Dickman Function

F(1/2) = A244009; F(1/3) = A175475; F(1/4) = A245238.

1/F(1/x) = 1/rho(x), where rho(x) satisfies rho'(x) = -rho(x-1)/x and rho(x) = 1 for x <= 1. rho(x) may be computed to arbitrary precision by the method of Marsaglia, Zaman and Marsaglia (implemented in the Python program in Links).

a(n) ~ exp(Ei(t) - n*t) / (t * sqrt(2*Pi*n)), where Ei is the exponential integral and t is the positive root of exp(t) - n*t - 1 (van de Lune and Wattel).

A344475 Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 20 2021

			0.10464776377316485385416972771819339482414269115729...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 286.

Pieter Moree, A special value of Dickman's function, Math. Student, Vol. 64 (1995), pp. 47-50; entire volume.
Pieter Moree, Nicolaas Govert de Bruijn, the enchanter of friable integers, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 774-801.
Eric Weisstein's World of Mathematics, Dickman Function.
Wikipedia, Dickman function.

Cf. A000796, A001622, A002390, A013661, A104457, A344476.

Cf. A175475, A244009 (value at 1/2), A245238, A309638.

RealDigits[1 - 2*Log[GoldenRatio] + Log[GoldenRatio]^2 - Pi^2/60, 10, 100][[1]]

my(phi = quadgen(5)); 1 - 2*log(phi) + log(phi)^2 - Pi^2/60 \\ Amiram Eldar, Jan 09 2025

Equals 1 - 2*log(phi) + log(phi)^2 - Pi^2/60 (Moree, 1995).

More terms from Amiram Eldar, Jan 09 2025

A291271 The arithmetic function v_4(n,2).

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 2, 4, 4, 5, 4, 6, 6, 7, 6, 8, 8, 9, 8, 10, 10, 11, 10, 12, 12, 13, 12, 14, 14, 15, 14, 16, 16, 17, 16, 18, 18, 19, 18, 20, 20, 21, 20, 22, 22, 23, 22, 24, 24, 25, 24, 26, 26, 27, 26, 28, 28, 29, 28, 30, 30, 31, 30, 32, 32, 33, 32, 34, 34

Offset: 2

Robert Price, Aug 21 2017

nonn

For any integer n>=7, a(n) is the smallest number of diametrical slices needed to divide two pizzas equally between n-4 people. - Jamil Silva, Mar 29 2025

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Cf. A244009, A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Cf. A291330.

seq((n-gcd(n,4))/2, n=2..80); # Ridouane Oudra, Dec 28 2024

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[4, n, 2], {n, 2, 70}]

Conjecture: a(n) = (n-2-cos(n*Pi)-cos(n*Pi/2))/2. - Wesley Ivan Hurt, Oct 02 2017
a(n) = (n-gcd(n,4))/2 = A291330(n)/2. - Ridouane Oudra, Dec 28 2024
Sum_{n>=5} (-1)^n/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Jan 15 2025
a(2)=a(4)=0, a(3)=1, a(5)=a(6)=2, a(2n+5)=n+2, a(4n+4)=2n, a(4n+6)=2n+2. - Jamil Silva, Mar 29 2025

A348371 Decimal expansion of Sum_{k>=0} binomial(2k,k)^2/(16^k(k+1)^3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 15 2021

			1.03928049679487622006025262010356644086601121330111...

Pablo Fernandez Refolio, Problem 12094, The American Mathematical Monthly, Vol. 126, No. 2 (2019), p. 180; Catalan's Constant from Catalan Numbers, Solution to Problem 12094 by Roberto Tauraso, ibid., Vol. 127, No. 8 (2020), pp. 755-757.

Cf. A000984, A006752, A143233, A244009.

RealDigits[48/Pi - 16*(1 - Log[2]) - 32*Catalan/Pi, 10, 100][[1]]

Equals 48/Pi - 16*(1 - log(2)) - 32*G/Pi, where G is Catalan's constant (A006752).

Equals 4F3(1/2, 1/2, 1, 1; 2, 2, 2; 1), where pFq() is the generalized hypergeometric function.

cp's OEIS Frontend

A377588 Decimal expansion of 7*zeta(3)/(2*Pi^2) - log(2) + 1/2.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A377589 Decimal expansion of 9*zeta(3)/(2*Pi^2) - log(2) + 1/3.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A377592 Decimal expansion of 9*zeta(3)/Pi^2 - 93*zeta(5)/(2*Pi^4) - log(2) + 1/4.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A263354 Decimal expansion of the generalized hypergeometric function 3F2(1/2,3/2,3/2; 5/2,5/2;x) at x=1/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Formula

A309638 Nearest integer to 1/F(1/x), where F(x) is the Dickman function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A344475 Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A291271 The arithmetic function v_4(n,2).

Views

A377588 Decimal expansion of 7zeta(3)/(2Pi^2) - log(2) + 1/2.

A377589 Decimal expansion of 9zeta(3)/(2Pi^2) - log(2) + 1/3.

A377592 Decimal expansion of 9zeta(3)/Pi^2 - 93zeta(5)/(2*Pi^4) - log(2) + 1/4.

A348371 Decimal expansion of Sum_{k>=0} binomial(2k,k)^2/(16^k(k+1)^3).