OFFSET
1,2
COMMENTS
First column of A027446. - Eric Desbiaux, Mar 29 2013
From Amiram Eldar and Thomas Ordowski, Aug 07 2019: (Start)
By Wolstenholme's theorem, if p > 3 is a prime, then p^2 | a(p-1).
Conjecture: for n > 3, if n^2 | a(n-1), then n is a prime.
Note that if n = p^2 with prime p > 3, then n | a(n-1).
It seems that composite numbers n such that n | a(n-1) are only the squares n = p^2 of primes p > 3.
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164.
The n-th triangular number n(n+1)/2 | a(n) for n = 1, 2, 6, 4422, ... (End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Frank A. Haight, and Robert B. Jones., "A probabilistic treatment of qualitative data with special reference to word association tests." Journal of Mathematical Psychology 11.3 (1974): 237-244. [Denominators of fractions in Eq. 21.] [Annotated scanned copy]
Frank A. Haight and N. J. A. Sloane, Correspondence, 1975
Yilmaz Simsek, Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions, arXiv:2108.10756 [math.NT], 2021.
FORMULA
a(n) = Sum_{k=1..n} lcm(1,2,...,n)/k. - Thomas Ordowski, Aug 07 2019
MAPLE
a:= n-> add(1/k, k=1..n)*ilcm($1..n):
seq(a(n), n=1..30); # Alois P. Heinz, Mar 14 2013
MATHEMATICA
Table[HarmonicNumber[n]*LCM @@ Range[n], {n, 27}] (* Arkadiusz Wesolowski, Mar 29 2012 *)
PROG
(GAP) List([1..30], n->Sum([1..n], k->1/k)*Lcm([1..n])); # Muniru A Asiru, Apr 02 2018
(PARI) a(n) = sum(k=1, n, 1/k)*lcm([1..n]); \\ Michel Marcus, Apr 02 2018
(Magma) [HarmonicNumber(n)*Lcm([1..n]):n in [1..30]]; // Marius A. Burtea, Aug 07 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved