A072345 -id:A072345 - cp's OEIS frontend

A072346 Volume of n-dimensional sphere of radius r is V_nr^n = Pi^(n/2)r^n/(n/2)! = C_nPi^floor(n/2)r^n; sequence gives denominator of C_n.

1, 1, 1, 3, 2, 15, 6, 105, 24, 945, 120, 10395, 720, 135135, 5040, 2027025, 40320, 34459425, 362880, 654729075, 3628800, 13749310575, 39916800, 316234143225, 479001600, 7905853580625, 6227020800, 213458046676875, 87178291200, 6190283353629375, 1307674368000

Offset: 0

N. J. A. Sloane, Jul 31 2002

Answer to question of how to extend the sequence 1, 2 r, Pi r^2, 4 Pi r^3 / 3, Pi^2 r^4 / 2, ...

Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1).

			Sequence of C_n's begins 1, 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, 1/24, 32/945, 1/120, 64/10395, ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9, Eq. 17.
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22; http://www.advancesindifferenceequations.com/content/2012/1/22. - From N. J. A. Sloane, Sep 04 2012

Eric Weisstein's World of Mathematics, Hypersphere
Eric Weisstein's World of Mathematics, Ball
Eric Weisstein's World of Mathematics, Four-Dimensional Geometry

Cf. A072345.

Cf. A001147.

f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Denominator[ f[n]], {n, 0, 30} ]

(n/2)! if n even, n!! if n odd.

A072478 Surface area of n-dimensional sphere of radius r is nV_nr^(n-1) = nPi^(n/2)r^(n-1)/(n/2)! = S_nPi^floor(n/2)r^(n-1); sequence gives numerator of S_n.

Original entry on oeis.org

0, 2, 2, 4, 2, 8, 1, 16, 1, 32, 1, 64, 1, 128, 1, 256, 1, 512, 1, 1024, 1, 2048, 1, 4096, 1, 8192, 1, 16384, 1, 32768, 1, 65536, 1, 131072, 1, 262144, 1, 524288, 1, 1048576, 1, 2097152, 1, 4194304, 1, 8388608, 1, 16777216, 1, 33554432, 1, 67108864, 1

Offset: 0

N. J. A. Sloane, Aug 02 2002

Answer to question of how to extend the sequence 0, 2, 2 Pi r, 4 Pi r^2, 2 Pi^2 r^3, ...
Volume of n-dimensional sphere of radius r is V_n*r^n - see A072345/A072346.
a(2*n-1) = 2^n and for n>2 a(2*n)=1.
Denominator of the rational coefficient of integral_{x>0} exp(-x^2)*x^n. - Jean-François Alcover, Apr 23 2013
From Ilya Gutkovskiy, Aug 02 2016: (Start)
Numerator of n/Gamma(n/2+1).
More generally, the ordinary generating function for the surface area of the n-dimensional sphere of radius r is 2*x*(1 + exp(Pi*r^2*x^2)*Pi*r*x + exp(Pi*r^2*x^2)*Pi*r*erf(sqrt(Pi)*r*x)*x) =  2*x + 2*Pi*r*x^2 + 4*Pi*r^2*x^3 + 2*Pi^2*r^3*x^4 + (8*Pi^2*r^4/3)*x^5 + Pi^3*r^5*x^6 + ... (End)

			Sequence of S_n's begins 0, 2, 2, 4, 2, 8/3, 1, 16/15, 1/3, 32/105, 1/12, 64/945, ...

N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 10, Eq. 19.

Colin Barker, Table of n, a(n) for n = 0..1000
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Ball
Eric Weisstein's World of Mathematics, Hypersphere
Eric Weisstein's World of Mathematics, Four-Dimensional Geometry
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A072479. A072478(n)/A072479(n) = n*A072345(n)/A072346(n).

f[n_] := Pi^(n/2 - Floor[n/2])*n/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 52}]
CoefficientList[Series[x (2 + 2 x - 2 x^2 - 4 x^3 - x^5 + 2 x^7)/(1 - 3 x^2 + 2 x^4), {x, 0, 52}], x] (* Michael De Vlieger, Aug 01 2016 *)
LinearRecurrence[{0,3,0,-2},{0,2,2,4,2,8,1,16,1},60] (* Harvey P. Dale, May 30 2018 *)

concat(0, Vec(x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7)/(1-3*x^2+2*x^4) + O(x^100))) \\ Colin Barker, Aug 01 2016

From Colin Barker, Sep 04 2012: (Start)
a(n) = 3*a(n-2)-2*a(n-4) for n>4.
G.f.: x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7) / (1-3*x^2+2*x^4).
(End)
From Colin Barker, Aug 01 2016: (Start)
a(n) = (1+(-1)^n-2^((1+n)/2)*(-1+(-1)^n))/2 for n>4.
a(n) = 1 for n>4 and even.
a(n) = 2^((n+1)/2) for n>4 and odd.
(End)

More terms from Robert G. Wilson v, Aug 18 2002

A072479 Surface area of n-dimensional sphere of radius r is nV_nr^(n-1) = nPi^(n/2)r^(n-1)/(n/2)! = S_nPi^floor(n/2)r^(n-1); sequence gives denominator of S_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 15, 3, 105, 12, 945, 60, 10395, 360, 135135, 2520, 2027025, 20160, 34459425, 181440, 654729075, 1814400, 13749310575, 19958400, 316234143225, 239500800, 7905853580625, 3113510400, 213458046676875, 43589145600

Offset: 0

N. J. A. Sloane, Aug 02 2002

Answer to question of how to extend the sequence 0, 2, 2 Pi r, 4 Pi r^2, 2 Pi^2 r^3, ...
Volume of n-dimensional sphere of radius r is V_n*r^n - see A072345/A072346.
Numerator of the rational coefficient of integral_{x>0} exp(-x^2)*x^n. [Jean-François Alcover, Apr 23 2013]

			Sequence of S_n's begins 0, 2, 2, 4, 2, 8/3, 1, 16/15, 1/3, 32/105, 1/12, 64/945, ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 10, Eq. 19.
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22; http://www.advancesindifferenceequations.com/content/2012/1/22 - From N. J. A. Sloane, Sep 04 2012

Eric Weisstein's World of Mathematics, Ball
Eric Weisstein's World of Mathematics, Hypersphere
Eric Weisstein's World of Mathematics, Four-Dimensional Geometry

Cf. A072478. A072478(n)/A072479(n) = n*A072345(n)/A072346(n).

f[n_] := Pi^(n/2 - Floor[n/2])*n/(n/2)!; Table[ Denominator[ f[n]], {n, 0, 30} ]

More terms from Robert G. Wilson v, Aug 18 2002

A074455 Consider the volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 22 2002

From David W. Wilson, Jul 12 2007: (Start)
For an integer d, the volume of a d-dimensional unit ball is v(d) = Pi^(d/2)/(d/2)! and its surface area is area(d) = d*Pi^(d/2)/(d/2)! = d*v(d). If we interpolate n! = gamma(n+1) we can define v(d) and area(d) as continuous functions for (at least) d >= 0.
A074457 purports to minimize area(d). Since area(d+2) = 2*Pi*v(d), area() is minimized at y = x+2; therefore A074457 coincides with the current sequence except at the first term. (End)

			5.256946404860576780132838388690769236619017237183214857509879678777109...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

Brian Hayes, An Adventure in the Nth Dimension, pp. 30-42 of M. Pitici, editor, The Best Writing on Mathematics 2012, Princeton Univ. Press, 2012. See p. 42. - From _N. J. A. Sloane_, Jan 13 2013; alternative link.
Eric Weisstein's World of Mathematics, Ball.

Cf. A074457.

The volume is given by A074454. Cf. A072345 & A072346.

x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)

hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
hyperspherevolume(d)=hyperspheresurface(d)/d
FindMax(fn_x,lo,hi)=
{
local(oldprecision, x, y, z);
oldprecision = default(realprecision);
default(realprecision, oldprecision+10);
while (hi-lo > 10^-oldprecision,
while (1,
z = vector(2, i, lo*(3-i)/3 + hi*i/3);
y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
default(realprecision, default(realprecision)+10);
);
if (y[1] < y[2], lo = z[1], hi = z[2]);
);
default(realprecision, oldprecision);
(lo + hi) / 2.
}
default(realprecision, 105);
A074455=FindMax("hyperspherevolume(x)", 1, 9)
A074457=FindMax("hyperspheresurface(x)", 1, 9)
A074454=hyperspherevolume(A074455)
A074456=hyperspheresurface(A074457)
/* David W. Cantrell */

2 * (solve(x=3, 4, psi(x) - log(Pi)) - 1) \\ Jianing Song, May 12 2025

d = root of Psi((1/2)*d + 1) = log(Pi).

d is 2 less than the number with decimal digits A074457 (the hypersphere dimension that maximizes hypersurface area). - Eric W. Weisstein, Dec 02 2014

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

A347045 Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

			The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The smallest divisor without the condition is A020639 (greatest: A006530).
Positions of 1's are A026424.
Positions of even terms are A063745 = 2*A026424.
The case of powers of 2 is A072345.
Positions of 2's are A100484.
Divisors of this type are counted by A345957 (rounded: A096825).
The rounded version is A347043.
The greatest divisor of this type is A347046 (rounded: A347044).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A001747, A027746, A038548, A056924, A063962, A106529, A140271, A244991, A324522, A335433, A335448.

Table[If[#=={},1,Min[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[1 ;; np/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[1:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347045(n):
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 1 if r else prod(fs[:q]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=1..A001222(n)/2} A027746(n,k) if A001222(n) is even, and 1 otherwise. - Amiram Eldar, Nov 02 2024

A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 3, 5, 1, 1, 1, 7, 5, 4, 1, 1, 1, 1, 7, 11, 1, 6, 5, 13, 1, 1, 1, 1, 1, 1, 11, 17, 7, 9, 1, 19, 13, 10, 1, 1, 1, 1, 1, 23, 1, 1, 7, 1, 17, 1, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 1, 8, 13, 1, 1, 1, 23, 1, 1, 1, 1, 37, 1, 1, 11, 1, 1, 1, 9

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

Problem: What are the positions of last appearances > 1?

			The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The greatest divisor without the condition is A006530 (smallest: A020639).
Positions of 1's are A026424.
The case of powers of 2 is A072345.
Positions of first appearances are A123667 (sorted: A123666).
Divisors of this type are counted by A345957 (rounded: A096825).
The rounded version is A347044.
The smallest divisor of this is A347045 (rounded: A347043).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A038548, A060775, A106529, A244991, A335433, A335448.

Table[If[#=={},1,Max[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[np/2+1 ;; np]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[-1:0:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347046(n):
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 1 if r else prod(fs[q:]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=A001222(n)/2+1..A001222(n)} A027746(n,k) if A001222(n) is even, and 1 otherwise. - Amiram Eldar, Nov 02 2024

A087299 Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

Original entry on oeis.org

1, 1, 4, 6, 32, 60, 384, 840, 6144, 15120, 122880, 332640, 2949120, 8648640, 82575360, 259459200, 2642411520, 8821612800, 95126814720, 335221286400, 3805072588800, 14079294028800, 167423193907200, 647647525324800

Offset: 0

Eric W. Weisstein, Aug 31 2003

			The volume of sphere (3-ball) is 4/3*Pi*r^3 and circumscribing 3-cube is 2^3*r^3 so ratio is Pi/6 and a(3)=6.
G.f. =  1 + x + 4*x^2 + 6*x^3 + 32*x^4 + 60*x^5 + 384*x^6 + 840*x^7 + ...

N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292

G. C. Greubel, Table of n, a(n) for n = 0..725
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Ball

Cf. A000407, A047053, A072345, A072346.

a[ n_] := If[ n < 0, 0, With[ {m = n + 1}, m! SeriesCoefficient[ Exp[x^2] (1 + Sqrt[Pi] Erf[x]), {x, 0, m}] / 2]]; (* Michael Somos, Jan 24 2014 *)
Table[2^n*Gamma[n/2 + 1]*Pi^Floor[n/2]/Pi^(n/2), {n,0,50}] (* G. C. Greubel, Jan 28 2017 *)

{a(n) = my(A); if( n<0, 0, n++; A = exp(x^2 + x * O(x^n)); n! * polcoeff( A * (1 + 2 * intformal( 1/A )), n) / 2)}; /* Michael Somos, May 25 2004 */

{a(n) = if( n<2, n>-1, 2*n * a(n-2))}; /* Michael Somos, Jan 24 2014 */

{a(n) = if( n<0, 0, if( n%2, n! / (n\2)!, 2^n * (n\2)!))}; /* Michael Somos, Jan 03 2015 */

a(n) = 2^n*gamma(n/2+1)*Pi^floor(n/2)/Pi^(n/2), n >= 0. - Wolfdieter Lang, Jul 17 2013
0 = a(n)*( 2*a(n+1) - a(n+3) ) + a(n+1)*a(n+2) if n>=0. - Michael Somos, Jan 24 2014
a(n) = 2*n * a(n-2) if n>=2. - Michael Somos, Jan 24 2014
a(2*n) = A047053(n). a(2*n + 1) = A000407(n). - Michael Somos, Jan 03 2015

A164103 Decimal expansion of 8*Pi^2/15.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 5-dimensional unit sphere.

For all nonnegative integers n, let V_n be the volume of the n-dimensional unit sphere. If n != 5, then V_n < V_5, this constant (see A072345). - Rick L. Shepherd, Feb 23 2014

			Equals 5.2637890139143245967117285332672806...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere.
Index entries for transcendental numbers.

RealDigits[8*Pi^2/15, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
8*Pi^2/15 \\ G. C. Greubel, Apr 09 2017
```

Equals A164104/5 = 4*A164102/15.

A351705 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the numerator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351706 for the denominators.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 5, 7, 1, 5, 3, 7, 9, 13, 11, 15, 4, 5, 9, 11, 6, 7, 13, 15, 17, 21, 19, 23, 25, 29, 27, 31, 1, 9, 5, 13, 17, 25, 21, 29, 3, 11, 7, 15, 19, 27, 23, 31, 33, 41, 37, 45, 49, 57, 53, 61, 35, 43, 39, 47, 51, 59, 55, 63, 8, 9, 17, 19, 10, 11, 21

Offset: 0

Rémy Sigrist, Feb 16 2022

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

			For n = 13:
- 13 = 2^0 + 2^2 + 2^3,
- A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,
- d(13) = 2^0 + 2^1 + 2^-2 = 13/4,
- so a(13) = 13.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Wikipedia, Dyadic rational
Index entries for sequences related to binary expansion of n

Cf. A000120, A000695, A072345, A351706, A351785, A351786.

a(n) = { my (d=0, k); while (n, n-=2^k=valuation(n,2); d+=2^((-1)^k*(k+1)\2)); numerator(d) }

a(A000695(n)) = n.
a(2^k) = A072345(k-1) for any k > 0.
a(2^k-1) = 2^k-1 for any k >= 0.
A000120(a(n)) = A000120(n).

A351706 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the denominator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351705 for the numerators.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 2, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 1, 2, 2, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 2, 2, 1, 1, 2

Offset: 0

Rémy Sigrist, Feb 16 2022

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

			For n = 13:
- 13 = 2^0 + 2^2 + 2^3,
- A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,
- d(13) = 2^0 + 2^1 + 2^-2 = 13/4,
- so a(13) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Wikipedia, Dyadic rational
Index entries for sequences related to binary expansion of n

Cf. A000695, A016116, A072345, A351705, A351785, A351786.

a(n) = { my (d=0, k); while (n, n-=2^k=valuation(n,2); d+=2^((-1)^k*(k+1)\2)); denominator(d) }

a(A000695(n)) = 1.
a(2^k) = A072345(k) for any k >= 0.
a(2^k-1) = A016116(k) for any k >= 0.

cp's OEIS Frontend

A072346 Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives denominator of C_n.

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A072478 Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1) = n*Pi^(n/2)*r^(n-1)/(n/2)! = S_n*Pi^floor(n/2)*r^(n-1); sequence gives numerator of S_n.

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A072479 Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1) = n*Pi^(n/2)*r^(n-1)/(n/2)! = S_n*Pi^floor(n/2)*r^(n-1); sequence gives denominator of S_n.

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A074455 Consider the volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

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A347045 Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

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A347046 Greatest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

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A087299 Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

A072346 Volume of n-dimensional sphere of radius r is V_nr^n = Pi^(n/2)r^n/(n/2)! = C_nPi^floor(n/2)r^n; sequence gives denominator of C_n.

A072478 Surface area of n-dimensional sphere of radius r is nV_nr^(n-1) = nPi^(n/2)r^(n-1)/(n/2)! = S_nPi^floor(n/2)r^(n-1); sequence gives numerator of S_n.

A072479 Surface area of n-dimensional sphere of radius r is nV_nr^(n-1) = nPi^(n/2)r^(n-1)/(n/2)! = S_nPi^floor(n/2)r^(n-1); sequence gives denominator of S_n.