A073002 -id:A073002 - cp's OEIS frontend

A086237 Decimal expansion of Porter's constant.

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

Offset: 1

Eric W. Weisstein, Jul 12 2003

In his 'Addendum' to his paper in the year 2000 Don Knuth writes: "Gustav Lochs deserves to be mentioned here, because his work preceded that of Porter by nearly 15 years and involved essentially the same constant. Perhaps we should [..] refer in future to the Lochs-Porter constant, instead of simply saying 'Porter's constant'." - Peter Luschny, Aug 26 2014

The average number of divisions required by the Euclidean algorithm, for a coprime pair of independently and uniformly chosen numbers in the range [1, N] is (12*log(2)/Pi^2) * log(N) + c + O(N^(e-1/6)), for any e>0, where c is this constant (Porter, 1975). - Amiram Eldar, Aug 27 2020

			1.4670780794339754728977984847072299534499033224148...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, p. 157
Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 113.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Donald E. Knuth, Evaluation of Porter's constant, Computers and Mathematics with Applications, Vol. 2, No. 2 (1976), pp. 137-139.
Gustav Lochs, Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmäßigen Kettenbrüche, Monatshefte für Mathematik, Vol. 65, No. 1 (1961), pp. 27-52, alternative link.
John William Porter, On a Theorem of Heilbronn, Mathematika, Vol. 22, No. 1 (1975), pp. 20-28.
Eric Weisstein's World of Mathematics, Porter's Constant.

Cf. A001620, A073002, A074962, A143304.

RealDigits[(6 Log[2] (48 Log[Glaisher] - Log[2] - 4 Log[Pi] - 2))/Pi^2 - 1/2, 10, 110][[1]] (* Eric W. Weisstein, Aug 22 2013 *)
RealDigits[(6 Log[2] (Pi^2 (-2 + 4 EulerGamma + Log[8]) - 24 Zeta'[2]))/Pi^4 - 1/2, 10, 110][[1]] (* Eric W. Weisstein, Aug 22 2013 *)

x=.25^default(realprecision)
(6*log(2)*(4-48*(zeta(-1+x)-zeta(-1))/x-log(2)-4*log(Pi)-2))/Pi^2 - 1/2 \\ Charles R Greathouse IV, Aug 22 2013

(6*log(2)*(4-48*zeta'(-1)-log(2)-4*log(Pi)-2))/Pi^2-1/2 \\ Charles R Greathouse IV, Dec 12 2013

6*log(2)/Pi^2*(3*log(2) + 4*Euler - 24/Pi^2*zeta'(2) - 2) - 1/2 \\ Michel Marcus, Aug 27 2014

Equals 6*(log(2)/Pi^2)*(3*log(2) + 4*Gamma -(24/Pi^2)*Zeta'(2) - 2) - 1/2.

Equals 6*log(2)*(48*log(A074962) - 4*log(Pi) - log(2) - 2)/Pi^2 - 1/2 (see Finch). - Stefano Spezia, Dec 01 2024

A340440 Decimal expansion of Sum_{k>=2} log(k)/(k^2-1).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 07 2021

			1.0231387264279392955...

R. J. Mathar, The series limit of sum_k 1/(k log k (log log k)^2), arXiv:0902.0789 [math.NA], 2009-2021, version 3, App. B.

Cf. A073002, A261506, A340485, A340484.

sumpos(k=2, log(k)/(k^2-1)) \\ Michel Marcus, Jan 09 2021

Equals Sum_{i>=1} -zeta'(2i) = A073002 + A261506 - Sum_{i>=3} zeta'(2i).
Sum_{k>=2} log(k)/(k^2-s) = -Sum_{i>=1} s^(i-1)*zeta'(2i) for |s|<4. - R. J. Mathar, May 03 2021
Equals log(2)/2 + Sum_{k>=1} (zeta(2*k)-1)/(2*k-1). - Amiram Eldar, Jun 08 2021

A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

Original entry on oeis.org

1, 5, 13, 25, 41, 65, 85, 121, 157, 205, 221, 325, 313, 425, 533, 569, 545, 785, 685, 1025, 1105, 1105, 1013, 1573, 1441, 1565, 1777, 2125, 1625, 2665, 1861, 2617, 2873, 2725, 3485, 3925, 2665, 3425, 4069, 4961, 3281, 5525, 3613, 5525, 6437, 5065, 4325, 7397, 5965

Offset: 1

Seiichi Manyama, May 21 2024

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Row sums of A350900.
Cf. A057660, A350156.
Cf. A001620, A002117, A013661, A073002, A244115, A306016.

f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)

a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2);} \\ Amiram Eldar, May 27 2024

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
From Amiram Eldar, May 27 2024: (Start)
Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A115522 Decimal expansion of (Glaisher^12/(2Pie^EulerGamma))^(Pi^2/6).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 25 2006

			2.5537126827482090529...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 135.

Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A000796, A001113, A001620, A073002, A074962.

RealDigits[(Glaisher^12/(2Pi E^EulerGamma))^(Pi^2/6),10,100][[1]] (* Vaclav Kotesovec, Aug 15 2015 after Eric W. Weisstein *)

Equals Product_{k>=1} k^(1/k^2). - Vaclav Kotesovec, Dec 10 2017
Equals (Product_{k>=1} prime(k)^(1/(prime(k)^2-1)))^(Pi^2/6) (Van Gorder, 2012). - Amiram Eldar, Jul 22 2022
Equals exp(-zeta'(2)). - Vaclav Kotesovec, Jun 22 2023

A129364 a(n) = Product_{k = 1..n} A066841(k).

Original entry on oeis.org

1, 2, 6, 96, 480, 207360, 1451520, 2972712960, 722369249280, 5778953994240000, 63568493936640000, 9111096278347394580480000, 118444251618516129546240000, 10400352846118664303196241920000

Offset: 1

Peter Bala, Apr 11 2007

Conjecture: a(n) divides A092287(n) for all n - see comments in A129365.

Cf. A007955, A066841, A092143, A092287, A129365.

Table[Product[Floor[n/k]!^k, {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 24 2021 *)
Table[Product[k^(Floor[n/k]*(1 + Floor[n/k])/2), {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 24 2021 *)

a(n) = prod(k=1, n, k^((n\k) * (1 + n\k) \ 2)); \\ Daniel Suteu, Sep 12 2018

a(n) = Product_{k = 1..n} Product_{d|k} d^(k/d).
a(n) = Product_{k = 1..n} ((floor(n/k))!)^k.
a(n) = exp(Sum_{k = 1..n} log(k)/2 * floor(n/k) * floor(1 + n/k)). - Daniel Suteu, Sep 12 2018
log(a(n)) ~ c * n^2, where c = -zeta'(2)/2 = A073002/2 = 0.468774... - Vaclav Kotesovec, Jun 24 2021

A174466 a(n) = Sum_{d|n} dsigma(n/d)tau(d).

Original entry on oeis.org

1, 7, 10, 31, 16, 70, 22, 111, 64, 112, 34, 310, 40, 154, 160, 351, 52, 448, 58, 496, 220, 238, 70, 1110, 166, 280, 334, 682, 88, 1120, 94, 1023, 340, 364, 352, 1984, 112, 406, 400, 1776, 124, 1540, 130, 1054, 1024, 490, 142, 3510, 316, 1162, 520, 1240

Offset: 1

Paul D. Hanna, Apr 04 2010

Compare to sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d) = sum of squares of divisors of n.
tau(n) = A000005(n) = the number of divisors of n,
and sigma(n) = A000203(n) = sum of divisors of n.
Dirichlet convolution of A038040 and A000203. - R. J. Mathar, Feb 06 2011

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

Cf. A000005 (tau), A000203 (sigma), A007425 (tau_3), A034718, A038040, A174465.

a174466 n = sum $ zipWith3 (((*) .) . (*))
                  divs (map a000203 $ reverse divs) (map a000005 divs)
                  where divs = a027750_row n
-- Reinhard Zumkeller, Jan 21 2014

[&+[d*DivisorSigma(1, n div d)*#Divisors(d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 18 2019

f[p_, e_] := ((e^2+3*e+2)*p^(e+3) - 2*(e^2+4*e+3)*p^(e+2) + (e^2+5*e+6)*p^(e+1) - 2)/(2*(p-1)^3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2025 *)

{a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d,0))}

Logarithmic derivative of A174465.
Dirichlet g.f.: zeta(s)*(zeta(s-1))^3. - R. J. Mathar, Feb 06 2011
a(n) = Sum_{d|n} tau_3(d)*d = Sum_{d|n} A007425(d)*d. - Enrique Pérez Herrero, Jan 17 2013
G.f.: Sum_{k>=1} k*tau_3(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018
Sum_{k=1..n} a(k) ~ Pi^2*n^2/24 * (log(n)^2 + ((6*g - 1) + 12*z1/Pi^2) * log(n) + (1 - 6*g + 12*g^2 - 12*sg1)/2 + 6*((6*g - 1)*z1 + z2)/Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(p^e) = ((e^2+3*e+2)*p^(e+3) - 2*(e^2+4*e+3)*p^(e+2) + (e^2+5*e+6)*p^(e+1) - 2)/(2*(p-1)^3). - Amiram Eldar, May 26 2025

A245074 Decimal expansion of B, the coefficient of n*log(n)^2 in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 11 2014

The coefficient of n*log(n)^3 in the same asymptotic formula is A = 1/Pi^2.

			0.744341276391456640439006036785694615691377808839427047585292094877364...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section Sierpinski's Constant, p. 124.

Adrian W. Dudek, An Elementary Proof of an Asymptotic Formula of Ramanujan, arXiv:1401.1514 [math.NT], 2014.
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).

Cf. A061502, A073002, A092742, A319090, A319091.

B = (12*EulerGamma - 3)/Pi^2 - (36/Pi^4)*Zeta'[2]; RealDigits[B, 10, 103] // First

(4*Euler-1)/(2*zeta(2)) - zeta'(2)/zeta(2)^2 \\ Amiram Eldar, Apr 27 2025

B = (12*gamma - 3)/Pi^2 - (36/Pi^4)*zeta'(2).

A319090 Decimal expansion of C, the coefficient of n*log(n) in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Sep 10 2018

			0.823265208269485020156816453947090406301273270321142250892524579208530395971755...

Ramanujan's Papers, Some formulas in the analytic theory of numbers Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).

Cf. A061502, A092742, A245074, A319091.

36*EulerGamma^2/Pi^2 - (288*Zeta'[2]/Pi^4 + 24/Pi^2)*EulerGamma + (864*Zeta'[2]^2/Pi^6 + 72*Zeta'[2]/Pi^4 - 72/Pi^4*Zeta''[2] + 6/Pi^2) - 24*StieltjesGamma[1]/Pi^2

C = 36*gamma^2/Pi^2 - (288*z1/Pi^4 + 24/Pi^2)*gamma + (864*z1^2/Pi^6 + 72*z1/Pi^4 - 72/Pi^4*z2 + 6/Pi^2) - 24*g1/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994 and g1 is the first Stieltjes constant, see A082633.

A319091 Decimal expansion of D, the coefficient of n in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Sep 10 2018

			0.4603233722587214303937620863844189747632149035387392240584250348445902629324...

Ramanujan's Papers, Some formulas in the analytic theory of numbers Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).

Cf. A061502, A092742, A245074, A319090.

24*EulerGamma^3/Pi^2 - (432*Zeta'[2] /Pi^4+ 36/Pi^2)*EulerGamma^2 + (3456*Zeta'[2]^2/Pi^6 + 288*(Zeta'[2]-Zeta''[2])/Pi^4 + 24/Pi^2 - 72*StieltjesGamma[1]/Pi^2)*EulerGamma + StieltjesGamma[1]*(288*Zeta'[2]/Pi^4 + 24/Pi^2)-10368*Zeta'[2]^3/Pi^8 - 864*Zeta'[2]^2/Pi^6 + 1728*Zeta''[2] * Zeta'[2]/Pi^6 + 72*(Zeta''[2]-Zeta'[2])/Pi^4 - 48*Zeta'''[2]/Pi^4 + (12*StieltjesGamma[2] - 6)/Pi^2

D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279.

A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

Original entry on oeis.org

1, 9, 21, 57, 77, 173, 201, 329, 410, 570, 614, 1046, 1098, 1322, 1562, 1962, 2030, 2678, 2754, 3474, 3810, 4162, 4254, 5790, 6015, 6431, 6863, 7871, 7987, 9907, 10031, 11183, 11711, 12255, 12815, 15731, 15879, 16487, 17111, 19671, 19835, 22523, 22695, 24279

Offset: 1

Vaclav Kotesovec, Oct 23 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Cf. A061502, A318755, A320897.

Cf. A000005, A006218, A143127, A319085, A320895.

Accumulate[Table[k*DivisorSigma[0, k]^2, {k, 1, 50}]]

a(n) = sum(k=1, n, k*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

a(n) ~ n^2 * (3*(Pi^6*(-1 - 24*g^2 + 32*g^3 + g*(8 - 96*s1) + 16*s1 + 16*s2) - 13824*z1^3 + 576*Pi^2*z1*((-1 + 8*g)*z1 + 4*z2) - 8*Pi^4*(3*(1 - 8*g + 24*g^2 - 16*s1)*z1 - 6*z2 + 48*g*z2 + 8*z3)) + 6*(Pi^6*(1 - 8*g + 24*g^2 - 16*s1) + 576*Pi^2*z1^2 - 24*Pi^4*(-z1 + 8*g*z1 + 2*z2))*log(n) + 6*((-1 + 8*g)*Pi^6 - 24*Pi^4*z1)*log(n)^2 + 4*Pi^6*log(n)^3) / (8*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.

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A086237 Decimal expansion of Porter's constant.

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A340440 Decimal expansion of Sum_{k>=2} log(k)/(k^2-1).

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A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

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A115522 Decimal expansion of (Glaisher^12/(2*Pi*e^EulerGamma))^(Pi^2/6).

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A129364 a(n) = Product_{k = 1..n} A066841(k).

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A174466 a(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

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A245074 Decimal expansion of B, the coefficient of n*log(n)^2 in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.

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A115522 Decimal expansion of (Glaisher^12/(2Pie^EulerGamma))^(Pi^2/6).

A174466 a(n) = Sum_{d|n} dsigma(n/d)tau(d).