A375522 - OEIS

%I #21 Oct 20 2024 23:41:36

%S 1,2,6,15,105,1155,1336335,892896284280,398631887241408183843480,

%T 19863422690705846097977473796903171171326157280,

%U 14091270035344566960604487534521565339065390839583445590118556137472614250693240040301050080

%N a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A375781(k)).

%C Let S(n) = Sum_{k = 1..n} 1 / (k*A375781(k)) = S1(n)/S2(n) when reduced to lowest terms, where S1(n) = A375521(n), S2(n) = the present sequence.

%C The differences S2(n) - S1(n) are surprisingly small: for n = 1,2,...,34 the values S2(n) - S1(n) are:

%C 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

%C suggesting the conjecture that they are always 1 except for n = 4 and 6 (compare the Theorem in A374983).

%H Alois P. Heinz, <a href="/A375522/b375522.txt">Table of n, a(n) for n = 0..13</a>

%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 The first few fractions are 0/1, 1/2, 5/6, 14/15, 103/105, 1154/1155, 1336333/1336335, 892896284279/892896284280, ...

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

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

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

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

%o (Python)

%o from itertools import islice

%o from math import gcd

%o from sympy import nextprime

%o def A375522_gen(): # generator of terms

%o     p, q, k = 0, 1, 1

%o     while (k:=nextprime(k)):

%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         yield q

%o A375522_list = list(islice(A375522_gen(),11)) # _Chai Wah Wu_, Aug 30 2024

%Y Cf. A375781, A375521.

%K nonn,frac

%O 0,2

%A _Rémy Sigrist_ and _N. J. A. Sloane_, Aug 30 2024

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 18 2024