|
%I #87 Sep 08 2022 08:44:49
%S 1,3,11,25,137,147,1089,2283,7129,7381,83711,86021,1145993,1171733,
%T 1195757,2436559,42142223,42822903,825887397,837527025,848612385,
%U 859193865,19994251455,20217344325,102157567401,103187226801,312536252003,315404588903,9227046511387
%N a(n) = (1/1 + 1/2 + ... + 1/n)*lcm{1,2,...,n}.
%C First column of A027446. - _Eric Desbiaux_, Mar 29 2013
%C From _Amiram Eldar_ and _Thomas Ordowski_, Aug 07 2019: (Start)
%C By Wolstenholme's theorem, if p > 3 is a prime, then p^2 | a(p-1).
%C Conjecture: for n > 3, if n^2 | a(n-1), then n is a prime.
%C Note that if n = p^2 with prime p > 3, then n | a(n-1).
%C It seems that composite numbers n such that n | a(n-1) are only the squares n = p^2 of primes p > 3.
%C Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164.
%C The n-th triangular number n(n+1)/2 | a(n) for n = 1, 2, 6, 4422, ... (End)
%H Alois P. Heinz, <a href="/A025529/b025529.txt">Table of n, a(n) for n = 1..1000</a>
%H Frank A. Haight, and Robert B. Jones., <a href="/A025529/a025529.pdf">"A probabilistic treatment of qualitative data with special reference to word association tests."</a> Journal of Mathematical Psychology 11.3 (1974): 237-244. [Denominators of fractions in Eq. 21.] [Annotated scanned copy]
%H Frank A. Haight and N. J. A. Sloane, <a href="/A002805/a002805.pdf">Correspondence, 1975</a>
%H Yilmaz Simsek, <a href="https://arxiv.org/abs/2108.10756">Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions</a>, arXiv:2108.10756 [math.NT], 2021.
%F a(n) = A001008(n)*A110566(n). - _Arkadiusz Wesolowski_, Mar 29 2012
%F a(n) = Sum_{k=1..n} lcm(1,2,...,n)/k. - _Thomas Ordowski_, Aug 07 2019
%p a:= n-> add(1/k, k=1..n)*ilcm($1..n):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Mar 14 2013
%t Table[HarmonicNumber[n]*LCM @@ Range[n], {n, 27}] (* _Arkadiusz Wesolowski_, Mar 29 2012 *)
%o (GAP) List([1..30],n->Sum([1..n],k->1/k)*Lcm([1..n])); # _Muniru A Asiru_, Apr 02 2018
%o (PARI) a(n) = sum(k=1, n, 1/k)*lcm([1..n]); \\ _Michel Marcus_, Apr 02 2018
%o (Magma) [HarmonicNumber(n)*Lcm([1..n]):n in [1..30]]; // _Marius A. Burtea_, Aug 07 2019
%Y Differs from A096617 at 7th term.
%Y Cf. A001008, A002805, A027446, A110566.
%K nonn
%O 1,2
%A _Clark Kimberling_
|
