A279859 a(n) is the integer r with |r| < prime(n)/2 such that (prime(n)*Sum_{k=1..prime(n)-1} (3*H(k-1)^2 + 4*H(k-1)/k)/(k^2*binomial(2k,k)) + 3*H(prime(n)-1)/prime(n)^2)/prime(n)^2 == r (mod prime(n)), where H(m) denotes the harmonic number Sum_{k=1..m} 1/k.

3, -1, 2, 6, 8, -11, 10, -14, 0, -16, 15, 14, -8, -26, -9, 29, -26, -27, -1, 37, -34, 47, 40, 20, -36, 26, 6, -57, 29, -55, -23, 53, -9, 58, 52, -65, 33, -37, -83, 3, 24, 73, -72, -66, -76, 105, 45, -108, 84, -84, 19, 109, -84, 21, -28, -19, -139

Offset

4,1

Comments

Conjecture: (i) We have the new identity

Sum_{k>0} (3*H(k-1)^2 + 4*H(k-1)/k)/(k^2*binomial(2k,k)) = Pi^4/360.

(ii) Let B_0, B_1, B_2, ... be the Bernoulli numbers. For any prime p > 3, we have the congruences

p*Sum_{k=1..p-1} (3*H(k-1)^2 + 4*H(k-1)/k)/(k^2*binomial(2k,k)) == -3*H(p-1)/p^2 - p^2/5*B_{p-5} (mod p^3)

and

Sum_{k=1..p-1} (3*H(k)^2 - 4*H(k)/k)*binomial(2k,k)/k

== 6*H(p-1)/p^2 + (8/5)*p^2*B_{p-5} (mod p^3).

It is known that H(p-1) == -(p^2/3)*B_{p-3} (mod p^3) for any prime p > 3. The first congruence in part (ii) of the conjecture implies that a(n) == -B_{prime(n)-5}/5 (mod prime(n)) for all n = 4,5,....

Links

Zhi-Wei Sun, Table of n, a(n) for n = 4..800

Zhi-Wei Sun,List of conjectural series for powers of pi and other constants, arXiv:1102.5649 [math.CA], 2011-2014.

Zhi-Wei Sun, New series for some special values of L-functions, Nanjing Univ. J. Math. Biquarterly 32(2015), no.2, 189-218.

Example

a(4) = 3 since prime(4) = 7 and (7*Sum_{k=1..6} (3*H(k-1)^2 + 4*H(k-1)/k)/(k^2*binomial(2k,k)) + 3*H(6)/7^2)/7^2 = 3 - 7*657251/1555200 == 3 (mod 7).

Mathematica

rMod[m_, n_]:=rMod[m, n]=Mod[Numerator[m]*PowerMod[Denominator[m], -1, n], n, -n/2];
R[n_]:=R[n]=rMod[(Prime[n]*Sum[(3*HarmonicNumber[k-1]^2+4*HarmonicNumber[k-1]/k)/(k^2*Binomial[2k, k]), {k, 1, Prime[n]-1}]+3*HarmonicNumber[Prime[n]-1]/Prime[n]^2)/Prime[n]^2, Prime[n]];
Table[R[n], {n, 4, 60}]

Crossrefs

Cf. A000040, A001008, A002805, A027641, A027642.

Sequence in context: A183289 A183253 A201675 * A201655 A049917 A308709

Adjacent sequences: A279856 A279857 A279858 * A279860 A279861 A279862

Keyword

sign

Author

Zhi-Wei Sun, Dec 20 2016

Status

approved