A256502 Largest integer not exceeding the harmonic mean of the first n squares.

1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41

Offset: 1

Author

Stanislav Sykora, Apr 08 2015

Keywords

• nonn

Comments

Least k such that 1/k <= mean of {1, 1/2^2, 1/3^2,..., 1/n^2}.

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A226762.

Programs

Mathematica

Table[Floor[HarmonicMean[Range[n]^2]],{n,70}] (* Harvey P. Dale, Mar 08 2020 *)

PARI

\\ Using only precision-independent integer operations:
a(n)=(n*n!^2)\sum(k=1,n,(n!\k)^2)

Formula

a(n) = floor(n/(Sum_{k=1..n} 1/k^2)).

Approaches asymptotically n/zeta(2), zeta being the Riemann function.

For any e > 0 and large enough n, n/zeta(2) + 36/Pi^4 - 1 < a(n) < n/zeta(2) + 36/Pi^4 + e. (Possibly this holds even with e = 0 for n > 29.) - Charles R Greathouse IV, Apr 08 2015