A330014 When prime(n) is an odd prime (n >= 2) and N(n) / D(n) = Sum_{k=1..prime(n)-1} 1/k^3, then prime(n) divides N(n) and a(n) = N(n) / prime(n).

3, 407, 4081, 1742192177, 1964289620189, 26430927136768997, 12913609418092462447, 14639800647032731764901, 21461951639001843544904995612963, 489697309796854053100609288112563213, 97796057728171000155497946604711651753457

Offset

2,1

Comments

The idea of this sequence comes from the 1st exercise of "sélection de la délégation française" in 2005 for IMO 2006 where it was asked to prove that prime(n) divides N(n) [See reference].

The first fractions N(n)/D(n) are 9/8, 2035/1728, 28567/24000, 19164113947/16003008000, 25535765062457/21300003648000, ...

References

Guy Alarcon and Yves Duval, TS: Préparation au Concours Général, RMS, Collection Excellence, Paris, 2010, chapitre 10, Exercices de sélection de la délégation française en Octobre 2005 pour OIM 2006, Exercice 1, p. 169, p. 179.

Links

Table of n, a(n) for n=2..12.

Index to sequences related to Olympiads.

Example

For prime(4) = 7 then 1 + 1/2^3 + 1/3^3 + 1/4^3 + 1/5^3 + 1/6^3 = 28567/24000 and 28567/7 = 4081, a(4) = 4081.

Mathematica

a[n_] := Numerator[Sum[1/(i- 1)^3, {i, 2, (p = Prime[n])}]]/p; Array[a, 11, 2] (* Amiram Eldar, Nov 27 2019 *)

Prog

(Magma) [(Numerator(&+ [1/(k-1)^3:k in [2..NthPrime(n)]])) / NthPrime(n):n in [2..12]]; // Marius A. Burtea, Nov 27 2019

Crossrefs

Cf. A076637, A061002, A076637 (Wolstenholme's Theorem).

Sequence in context: A203563 A181990 A198666 * A305664 A152517 A333134

Adjacent sequences: A330011 A330012 A330013 * A330015 A330016 A330017

Keyword

nonn

Author

Bernard Schott, Nov 27 2019

Status

approved