A374663 -id:A374663 - cp's OEIS frontend

A375518 First differences of A375516.

1, 2, 8, 36, 1152, 240000, 9696240000, 11752718457744180000, 15347376819435640459450549232576160000, 23554197523775043569951631809272942030408567274885169881327076295276944000

Offset: 0

N. J. A. Sloane, Aug 25 2024

nonn

The terms of A375516 are not well-understood. The present sequence was suggested by the fact that, from a certain point on, the terms of A375516 end with the digits 1200. If powers of 2 and 3 are ignored, the terms of the present sequence appear to be perfect squares.

Alois P. Heinz, Table of n, a(n) for n = 0..12

Cf. A374663, A374983, A375516, A375517.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:
b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:
a:= n-> denom(s(n+1))-denom(s(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 19 2024

s[n_] := s[n] = If[n == 0, 0, s[n - 1] + 1/(n*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*n)];
a[n_] := Denominator[s[n + 1]] - Denominator[s[n]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 14 2025, after Alois P. Heinz *)

a(0)=1 prepended by Alois P. Heinz, Oct 19 2024

A375520 a(n) = A375516(n)/LCM{1,...,n}.

Original entry on oeis.org

1, 2, 2, 2, 4, 20, 4020, 23086860, 13991331508857930, 6090228896601444631429868134927310, 9346903779275810940456996749711484938792041307270162838305692061624510

Offset: 0

N. J. A. Sloane, Aug 28 2024

nonn

It is a theorem of Rémy Sigrist (see the proof in A374983) that a(n) is an integer.

Alois P. Heinz, Table of n, a(n) for n = 0..14

Cf. A003418, A374663, A374983, A375516.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:
b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:
a:= n-> denom(s(n))/ilcm($1..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 19 2024

s[n_] := s[n] = If[n == 0, 0, s[n - 1] + 1/(n*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*n)];
a[n_] := If[n == 0, 1, Denominator[s[n]]/LCM @@ Range[n]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 14 2025, after Alois P. Heinz *)

from itertools import count, islice
from math import gcd, lcm
def A375520_gen(): # generator of terms
    p, q, c = 0, 1, 1
    for k in count(1):
        m = q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
        c = lcm(c,k)
        yield q//c
A375520_list = list(islice(A375520_gen(),11)) # Chai Wah Wu, Aug 28 2024

a(0)=1 prepended by Alois P. Heinz, Oct 19 2024

A375527 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (A000959(k)*a(k)) < 1 (where A000959(k) is the k-th lucky number).

Original entry on oeis.org

2, 1, 1, 5, 49, 3823, 10436791, 91498340590348, 16878924054006628616561542268, 1037072167459498271969377959736955928500322755810409274896, 1758618383011028875762229897498966705737981284604676043205492817705756616240608451710873787593445097075800445688725

Offset: 1

N. J. A. Sloane, Sep 01 2024

nonn

Exact analog of A375781, with the primes replaced by the lucky numbers (A000959).

The motivation was to see if the unusual properties of the partial sums arising from A375781 and from A374663 would hold for other divergent series. It appears that they certainly hold here - see A375528.

N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A000959, A374663, A375781, A375528.

A375528 a(n) = denominator of Sum_{k = 1..n} 1 / (A000959(k)*A375527(k)).

Original entry on oeis.org

1, 2, 6, 42, 630, 57330, 219172590, 2287458514758690, 523246645674205487113407810300, 34223381526163442974989472671319545640510650941743506071550, 65068880171408068403202506207461768112305307530373013598603234255112994800902512713302330140957468591804616490482800

Offset: 1

N. J. A. Sloane, Sep 01 2024

nonn

The first few sums S(n) = Sum_{k = 1..n} 1/(A000959(k)*A375527(k)) are: 1/2, 5/6, 41/42, 629/630, 57329/57330,
 219172589/219172590, 2287458514758689/2287458514758690,
 523246645674205487113407810299/523246645674205487113407810300, ..., and the first 10 or 11 of these sums have the form (c-1)/c, where c is an integer. The present sequence gives the denominators.
For the harmonic series analog, A374663, Rémy Sigrist has shown that all the partial sums have that form (see A374983), and for the prime number analog, A375581, it seems that all partial sums except for n = 4 and 6 have this property (see A375521/A375522).

N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A000959, A375527, A374683, A374983/A375516, A375582, A375521/A375522.

A376060 Lexicographically earliest sequence of positive integers a(0), a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 0..n-1} Catalan(k)/a(k) < 1.

Original entry on oeis.org

2, 3, 13, 391, 426973, 546916547269, 940084230410591812263433, 2872214670866692695441731060944339347071024216683

Offset: 0

N. J. A. Sloane, Sep 14 2024

Cf. A000108, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376059, A376061.

a(n+1) = Catalan(n+1)*A376061(n) + 1.

cp's OEIS Frontend

A375518 First differences of A375516.

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A375520 a(n) = A375516(n)/LCM{1,...,n}.

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A375527 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (A000959(k)*a(k)) < 1 (where A000959(k) is the k-th lucky number).

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A375528 a(n) = denominator of Sum_{k = 1..n} 1 / (A000959(k)*A375527(k)).

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A376060 Lexicographically earliest sequence of positive integers a(0), a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 0..n-1} Catalan(k)/a(k) < 1.

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