login
a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A374663(k)).
12

%I #47 Oct 20 2024 23:39:15

%S 0,1,3,11,47,1199,241199,9696481199,11752718467440661199,

%T 15347376819435640471203267700016821199,

%U 23554197523775043569951631809272942045755944094320810352530343995293765199

%N a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A374663(k)).

%C For the denominators see A375516 and A375517.

%C For n = 1..36, Sum_{k = 1..n} 1 / (k*A374663(k)) = a(n) / (1 + a(n)). In fact this holds for all n >= 1.

%C Theorem: Let S_n = Sum_{k = 1..n} 1 / (k*A374663(k)) and let r_n = 1 - S_n. Then for n > 1, r_n is the inverse of a positive integer, say d_n; d_{n+1} is divisible by d_n; and d_n is divisible by all positive integers < n. (See Sigrist link for proof; d_n is given in A375516.)

%H N. J. A. Sloane, <a href="/A374983/b374983.txt">Table of n, a(n) for n = 0..14</a>

%H Rémy Sigrist, <a href="/A374983/a374983.txt">Proof of Theorem</a>, Aug 26 2024, revised Sep 01 2024.

%H N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=3RAYoaKMckM">A Nasty Surprise in a Sequence and Other OEIS Stories</a>, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane85BD.pdf">Slides</a> [Mentions this sequence]

%e For n = 3: A374663(1) = A374663(2) = A374663(3) = 2, 1/(1*2) + 1/(2*2) + 1/(3*2) = 11/12, so a(3) = 11.

%p s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:

%p b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:

%p a:= n-> numer(s(n)):

%p seq(a(n), n=0..10); # _Alois P. Heinz_, Oct 18 2024

%o (PARI) { print1 (0); t = 0; for (n = 1, 10, for (v = c=ceil(1/(n*(1-t))), oo, if (t + 1/(n*v) < 1, t += 1/(n*v); print1 (", " numerator(t)); break;););); }

%o (Python)

%o from itertools import count, islice

%o from math import gcd

%o def A374983_gen(): # generator of terms

%o p, q = 0, 1

%o for k in count(1):

%o yield p

%o m = q//(k*(q-p))+1

%o p, q = p*k*m+q, k*m*q

%o p //= (r:=gcd(p,q))

%o q //= r

%o A374983_list = list(islice(A374983_gen(),11)) # _Chai Wah Wu_, Aug 28 2024

%Y Cf. A374663, A375516 (denominators), A375517.

%K nonn,frac

%O 0,3

%A _Rémy Sigrist_, Aug 04 2024