cp's OEIS Frontend

a(n) ~ n * (A1*log(n)^7 + A2*log(n)^6 + A3*log(n)^5 + A4*log(n)^4 + A5*log(n)^3 + A6*log(n)^2 + A7*log(n) + A8) [Ramanujan, 1916, formula (8)].

From Vaclav Kotesovec, Mar 12 2023: (Start)

Let f(s) = Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), then

A1 = f(1)/5040 = 0.0000097860463451190658257888710490039661018239924009134296302566263529129...

A2 = ((8*gamma - 1)*f(1) + f'(1)) / 720 = 0.0007019997226174095261771358653540021199703406583347258622085873074052900...

A3 = (2 * f(1) * (1 - 8*gamma + 28*gamma^2 - 8*sg1) + 2*(8*gamma - 1)*f'(1) + f''(1)) / 240 = 0.0171707557268638504150726777646428533953516776541779590118582753709080243...

A4 = (6*f(1)*(-1 - 28*gamma^2 + 56*gamma^3 + gamma*(8 - 56*sg1) + 8*sg1 + 4*sg2) + 6*(1 - 8*gamma + 28*gamma^2 - 8*sg1)*f'(1) + (24*gamma - 3)*f''(1) + f'''(1)) / 144 = 0.1758477246705824231478998937203303065702508974398264386862202155788...,

where f(1) = Product_{p prime} (1 - 9/p^2 + 16/p^3 - 9/p^4 + 1/p^6) = 0.0493216735794000917619759100869799891531929217006036853364933968186814900...,

f'(1) = f(1) * Sum_{p prime} 6*(3*p + 1) * log(p) / ((p-1) * (p^2 + 4*p + 1)) = 0.3270075329904166293296173488834535949530448497141635531152019426434776932...,

f''(1) = f'(1)^2 / f(1) + f(1) * Sum_{p prime} -36 * p^2 * (p+1)^2 * log(p)^2 / ((p-1)^2 * (p^2 + 4*p + 1)^2) = 1.1340946589859924227356699847227569935993284591079455746283572890834872890...,

f'''(1) = 3*f'(1)*f''(1)/f(1) - 2*f'(1)^3/f(1)^2 + f(1) * Sum_{p prime} 72*p^2 * (p^5 + 3*p^4 + 8*p^3 + 8*p^2 + 3*p+ 1) * log(p)^3 / ((p-1)^3 * (p^2+ 4*p + 1)^3) = -1.3447542210274297874241826540796632006263184659735145444999327537246287...,

gamma is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279.

Approximate values of other constants:

A5 = 0.7626157870664479996781152281270580148665443022014605423466363134512...

A6 = 1.3720912878905940866975369743071441424192833481004753922122458993040...

A7 = 1.1416118168318711437057727816148048057614284471759625288073915723140...

A8 = 0.2618221765943171424958051160111945242076019991649774700610674747694...

(End)

a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*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.