A086463 -id:A086463 - cp's OEIS frontend

A112102 Numerator of Sum_{i=1..n} 1/(i^3C(2i,i)).

0, 1, 25, 1129, 63251, 1581371, 52185743, 33242372291, 24176277773, 40688677687159, 2378722720177, 3741730846458901, 86059809503772983, 72720539036773885963, 72720539038037143387, 52722390802769505531767, 13075152919096992777263341

Offset: 0

N. J. A. Sloane, Nov 30 2005

			0, 1/2, 25/48, 1129/2160, 63251/120960, 1581371/3024000, 52185743/99792000, ... -> Pi^2/18.

Robert Israel, Table of n, a(n) for n = 0..576
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

Cf. A086463 (Pi^2/18), A112103 (denominator).

f:= proc(n) local i; numer(add(1/(i^3*binomial(2*i,i)),i=1..n)) end proc:
map(f, [$0..20]); # Robert Israel, Jun 22 2023

Join[{0},Numerator[Accumulate[Table[1/(i^3 Binomial[2i,i]),{i,20}]]]] (* Harvey P. Dale, May 28 2014 *)

a(n) = numerator(sum(i=1, n, 1/(i^3*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

Definition corrected (and an incorrect sum deleted) by Wolfdieter Lang, Oct 07 2008

A112103 Denominator of Sum_{i=1..n} 1/(i^3C(2i,i)).

Original entry on oeis.org

1, 2, 48, 2160, 120960, 3024000, 99792000, 63567504000, 46230912000, 77806624896000, 4548694993920, 7155097225436160, 164567236185031680, 139059314576351769600, 139059314576351769600, 100818003067855032960000, 25002864760828048174080000

Offset: 0

N. J. A. Sloane, Nov 30 2005

			0, 1/2, 25/48, 1129/2160, 63251/120960, 1581371/3024000, 52185743/99792000, ... -> Pi^2/18.

Robert Israel, Table of n, a(n) for n = 0..576
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

Cf. A086463 (Pi^2/18), A112102 (numerator).

f:= proc(n) local i; denom(add(1/(i^3*binomial(2*i,i)),i=1..n)) end proc:
map(f, [$0..20]); # Robert Israel, Jun 22 2023

Table[Sum[1/(k^3 Binomial[2k,k]),{k,n}],{n,0,20}]//Denominator (* Harvey P. Dale, Feb 19 2023 *)

a(n) = denominator(sum(i=1, n, 1/(i^3*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

Definition corrected (and an incorrect sum deleted) by Wolfdieter Lang, Oct 07 2008

A280098 The sum of the divisors of 24*n - 1, divided by 24.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 7, 8, 11, 10, 11, 14, 13, 17, 15, 16, 19, 18, 28, 20, 21, 24, 25, 31, 25, 30, 27, 31, 35, 30, 31, 35, 38, 41, 35, 36, 37, 38, 54, 46, 41, 45, 43, 53, 49, 46, 57, 48, 62, 55, 51, 55, 56, 76, 55, 60, 57, 63, 71, 60, 80, 62, 63, 77, 65, 66, 67

Offset: 1

Michael Somos, Dec 25 2016

Conjecture: only the integers k in {1, 3, 4, 6, 8, 12, 24} have the property that the sum of the divisors of (k*n-1)/k is always an integer. - Robert G. Wilson v, Dec 25 2016

The finite sequence mentioned in the above conjecture gives the sum of the divisors of the partition numbers of the first seven positive integers (cf. A139041). - Omar E. Pol, Dec 25 2016

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 7*x^7 + 8*x^8 + 11*x^9 + ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A086463, A139041, A183010, A280097.

a[ n_] := If[ n < 1, 0, DivisorSigma[ 1, 24 n - 1] / 24];
DivisorSigma[1,24*Range[70]-1]/24 (* Harvey P. Dale, Sep 25 2017 *)

{a(n) = if( n<1, 0, sigma(24*n - 1) / 24)};

24 * a(n) = sum of the divisors of A183010(n).
a(n) = A280097(n)/24. - Omar E. Pol, Dec 25 2016
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

A082050 Sum of divisors of n that are not of the form 3k+1.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 0, 10, 12, 7, 11, 23, 0, 16, 23, 10, 17, 38, 0, 27, 24, 13, 23, 55, 5, 28, 39, 16, 29, 61, 0, 42, 47, 19, 40, 86, 0, 40, 42, 35, 41, 88, 0, 57, 77, 25, 47, 103, 0, 57, 71, 28, 53, 119, 16, 80, 60, 31, 59, 153, 0, 64, 96, 42, 70, 121, 0, 87, 95, 56, 71, 190

Offset: 1

Ralf Stephan, Apr 02 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A004611, A027748, A046913, A078181, A078182, A082051, A086463.

sd[n_]:=Total[Select[Divisors[n],!IntegerQ[(#-1)/3]&]]; Array[sd,80] (* Harvey P. Dale, May 04 2011 *)

for(n=1,100,print1(sumdiv(n,d,if(d%3!=1,d))","))

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, (3*n-1)*x^(3*n-1)/(1-x^(3*n-1)) + (3*n)*x^(3*n)/(1-x^(3*n)) );
v = Vec(gf);  concat([0],v)
\\ Joerg Arndt, May 17 2013

a(A004611(n)) = 0.
G.f.: Sum_{k>=1} x^(2*k)*(2+3*x^k+x^(3*k))/(1-x^(3*k))^2. - Vladeta Jovovic, Apr 11 2006
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Jan 06 2024

A082051 Sum of divisors of n that are not of the form 3k+2.

Original entry on oeis.org

1, 1, 4, 5, 1, 10, 8, 5, 13, 11, 1, 26, 14, 8, 19, 21, 1, 37, 20, 15, 32, 23, 1, 50, 26, 14, 40, 40, 1, 65, 32, 21, 37, 35, 8, 89, 38, 20, 56, 55, 1, 80, 44, 27, 73, 47, 1, 114, 57, 36, 55, 70, 1, 118, 56, 40, 80, 59, 1, 141, 62, 32, 104, 85, 14, 131, 68, 39, 73, 88, 1, 185

Offset: 1

Ralf Stephan, Apr 02 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A003627, A027748, A046913, A078181, A078182, A082050, A086463, A326394.

sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 2) / 3]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)

for(n=1,100,print1(sumdiv(n,d,if(d%3!=2,d))","))

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, (3*n-2)*x^(3*n-2)/(1-x^(3*n-2)) + (3*n)*x^(3*n)/(1-x^(3*n)) );
v = Vec(gf)
\\ Joerg Arndt, May 17 2013

a(A003627(n)) = 1.
G.f.: Sum_{k>=1} x^k*(1 + 3*x^(2*k) + 2*x^(3*k))/(1 - x^(3*k))^2. - Ilya Gutkovskiy, Sep 12 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Jan 06 2024

A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.5054294746835192416424504819084321491886690145682628649826647...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)).

S[5] = 2*Pi*Im[PolyLog[4, (-1)^(1/3)]] + (1/9)*Pi^2*Zeta[3] - 19*Zeta[5]/3; RealDigits[S[5], 10, 103] // First

suminf(n=1, 1/(n^5*binomial(2*n,n))) \\ Michel Marcus, Sep 03 2015

S(5) = 2*Pi*Im(PolyLog(4, (-1)^(1/3))) + (1/9)*Pi^2*zeta(3) -19*zeta(5)/3.
Equals (1/2) 4F3(1,1,1,1; 3/2,2,2; 1/4).
Also equals (1/(2592*sqrt(3)))*(Pi*(PolyGamma(3, 1/6) + PolyGamma(3, 1/3) - PolyGamma(3, 2/3) - PolyGamma(3, 5/6))) + (1/9)*Pi^2*zeta(3) - 19*zeta(5)/3.

A264740 Sum of odd parts of divisors of n.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 8, 4, 13, 12, 12, 12, 14, 16, 24, 5, 18, 26, 20, 18, 32, 24, 24, 16, 31, 28, 40, 24, 30, 48, 32, 6, 48, 36, 48, 39, 38, 40, 56, 24, 42, 64, 44, 36, 78, 48, 48, 20, 57, 62, 72, 42, 54, 80, 72, 32, 80, 60, 60, 72, 62, 64, 104, 7

Offset: 1

Franklin T. Adams-Watters, Nov 22 2015

It is easy to show that a(n) is odd iff n is a square.
a(n) = sigma(n) for odd n, since any divisor of an odd number is odd.
Inverse Möbius transform of A000265(n). - Wesley Ivan Hurt, Jun 26 2025

			Divisors of 10 are 1, 2, 5, 10. The odd parts of these are 1, 1, 5, 5, so a(10) = 1+1+5+5 = 12.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000593, A000265, A000203.

Cf. A027750, A086463.

a264740 = sum . map a000265 . a027750_row'
-- Reinhard Zumkeller, Nov 23 2015

with(numtheory): with(padic): seq(add(d/2^ordp(d,2), d in divisors(n)), n=1..80); # Ridouane Oudra, Oct 30 2023

f[p_, e_] := If[p == 2, e + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 30 2020 *)

a(n)=my(k=valuation(n,2));sigma(n)\(2^(k+1)-1)*(k+1)

Multiplicative with a(2^k) = k + 1, a(p^k) = sigma(p^k) = (p^(k+1)-1) / (p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Nov 04 2022
a(n) = Sum_{d|n} A000265(d). - Ridouane Oudra, Oct 30 2023

A214361 Expansion of c(q^2) * (c(q) + 2 * c(q^4)) / 9 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 3, 3, 6, 9, 8, 3, 9, 18, 12, 9, 14, 24, 18, 3, 18, 27, 20, 18, 24, 36, 24, 9, 31, 42, 27, 24, 30, 54, 32, 3, 36, 54, 48, 27, 38, 60, 42, 18, 42, 72, 44, 36, 54, 72, 48, 9, 57, 93, 54, 42, 54, 81, 72, 24, 60, 90, 60, 54, 62, 96, 72, 3, 84, 108, 68, 54

Offset: 1

Michael Somos, Jul 18 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

b(n) = 6*a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z is not divisible by 3. - Michael Somos, Jun 23 2018

			G.f. = q + 3*q^2 + 3*q^3 + 3*q^4 + 6*q^5 + 9*q^6 + 8*q^7 + 3*q^8 + 9*q^9 + 18*q^10 + ...
a(1) = 1, b(1) = 6 with solutions (w, x, y, z) = {(0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)} and their negatives. - _Michael Somos_, Jun 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A086463, A121443, A124449, A185717, A214456.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - EllipticTheta[ 3, 0, q^3]^4) / 8, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(3/2)]^4 / (-16 (-q)^(1/2)), {q, 0, n}];

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};

{a(n) = if( n<1, 0, sigma(n) + if( n%3==0, -1 * sigma(n/3)) + if( n%4==0, -4 * sigma(n/4)) + if( n%12==0, +4 * sigma(n/12)))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 3, p==3, 3^e, (p^(e+1) - 1) / (p - 1))))};

Expansion of (phi(q)^4 - phi(q^3)^4) / 8 = q * phi(q^3) * (chi(q) * psi(-q^3))^3 = q * psi(-q^3)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2)^6 * eta(q^3) * eta(q^6)^2 * eta(q^12) / ( eta(q) * eta(q^4))^3 in powers of q.
Euler transform of period 12 sequence [3, -3, 2, 0, 3, -6, 3, 0, 2, -3, 3, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A214456.
a(n) is multiplicative with a(2^e) = 3 if e>0, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f.: x * Product_{k>0} (1 + (-x)^(3*k)) * (1 - x^(6*k))^4 / ( 1 + (-x)^k)^3.
a(n) = -(-1)^n * A124449(n). a(3*n) = 3*a(n). a(2*n) = a(4*n) = 3 * A121443(n). a(2*n + 1) = A185717(n).
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^(2*s-2)) * (1 - 1/3^s) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). (End)

A261850 Decimal expansion of the central binomial sum S(6), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.50267652147826928645467745997934863966460260009164...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261851 (S(7)), A261852 (S(8)).

S[6] = Sum[1/(n^6*Binomial[2n, n]), {n, 1, Infinity}]; RealDigits[S[6], 10, 105]//First

Equals (1/2) 7F6(1,1,1,1,1,1,1; 3/2,2,2,2,2,2; 1/4).

Also equals (2/3)*Integral_{0..Pi/3} t*log(2*sin(t/2))^4 dt.

A261851 Decimal expansion of the central binomial sum S(7), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.501325872688178809402296710552749443726878329858...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261852 (S(8)).

S[7]=-6*Pi*Im[-PolyLog[6, (-1)^(1/3)]] + (17*Pi^4*Zeta[3])/1620 + (1/3)*Pi^2*Zeta[5] - (493*Zeta[7])/24; RealDigits[S[7], 10, 105]//First

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals -6*Pi*Im(-PolyLog(6, (-1)^(1/3))) + (17*Pi^4*zeta(3))/1620 + (1/3)*Pi^2*zeta(5) - (493*zeta(7))/24.

cp's OEIS Frontend

A112102 Numerator of Sum_{i=1..n} 1/(i^3*C(2*i,i)).

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A112103 Denominator of Sum_{i=1..n} 1/(i^3*C(2*i,i)).

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A280098 The sum of the divisors of 24*n - 1, divided by 24.

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A082050 Sum of divisors of n that are not of the form 3k+1.

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A082051 Sum of divisors of n that are not of the form 3k+2.

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A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).

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A264740 Sum of odd parts of divisors of n.

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A112102 Numerator of Sum_{i=1..n} 1/(i^3C(2i,i)).

A112103 Denominator of Sum_{i=1..n} 1/(i^3C(2i,i)).