A375516 -id:A375516 - cp's OEIS frontend

A375517 a(n) = A375516(n)/n.

2, 2, 4, 12, 240, 40200, 1385211600, 1469089808430082650, 1705264091048404496800363077779646800, 2355419752377504356995163180927294204575594409432081035253034399529376520

Offset: 1

N. J. A. Sloane, Aug 20 2024

nonn

			The prime factors (without repetition) of the first ten terms are:
  {2},
  {2},
  {2},
  {2, 3},
  {2, 3, 5},
  {2, 3, 5, 67},
  {2, 3, 5, 67, 5743},
  {2, 3, 5, 7, 67, 5743, 1212060151},
  {2, 5, 7, 67, 137, 151, 5743, 10867, 1212060151, 5808829669},
  {2, 3, 5, 7, 19, 47, 67, 71, 137, 151, 5743, 10867, 1212060151, 5808829669, 243254025696427, 99509446928973841}

Alois P. Heinz, Table of n, a(n) for n = 1..14

Cf. 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))/n:
seq(a(n), n=1..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]]/n;
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Mar 21 2025, after Alois P. Heinz *)

from itertools import count, islice
from math import gcd
def A375517_gen(): # generator of terms
    p, q = 0, 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
        yield q//k
A375517_list = list(islice(A375517_gen(),11)) # Chai Wah Wu, Aug 28 2024

A375791 a(n) = A375516(n+1) / A375516(n).

Original entry on oeis.org

2, 2, 3, 4, 25, 201, 40201, 1212060151, 1305857607493406801, 1534737681943564047120326770001682121, 11777098761887521784975815904636471022877972047160405176265171997646882601

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Aug 29 2024

nonn

			a(7) = A375516(8) / A375516(7) = 11752718467440661200 / 9696481200 = 1212060151.

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

Cf. A374663, 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+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, 10}] (* Jean-François Alcover, Jul 02 2025, after Alois P. Heinz *)

{ r = 1; for (n = 1, 11, a = floor(1/(r*n))+1; d = denominator(r); r -= 1/(n*a); print1 (denominator(r)/d", ");); }

from itertools import count, islice
from math import gcd
def A375791_gen(): # generator of terms
    p, q = 0, 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
        yield k*m//r
A375791_list = list(islice(A375791_gen(),11)) # Chai Wah Wu, Aug 30 2024

A375518 First differences of A375516.

Original entry on oeis.org

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

A374663 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (k*a(k)) < 1.

Original entry on oeis.org

2, 2, 2, 4, 10, 201, 34458, 1212060151, 1305857607493406801, 1534737681943564047120326770001682121, 2141290683979549415450148346297540185977813099483710032048213090481251382

Offset: 1

Rémy Sigrist, Aug 04 2024

nonn

The harmonic series, Sum_{k > 0} 1/k, diverges. We divide each of its terms in such a way as to have a series bounded by 1.

			The initial terms, alongside the corresponding sums, are:
  n  a(n)        Sum_{k=1..n} 1/(k*a(k))
  -  ----------  -----------------------------------------
  1           2  1/2
  2           2  3/4
  3           2  11/12
  4           4  47/48
  5          10  1199/1200
  6         201  241199/241200
  7       34458  9696481199/9696481200
  8  1212060151  11752718467440661199/11752718467440661200
...
The denominators are in A375516.

Rémy Sigrist and N. J. A. Sloane, Dampening Down a Divergent Series, Manuscript in preparation, September 2024.

N. J. A. Sloane, Table of n, a(n) for n = 1..14
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. A000058, A001008/A002805, A002387, A374983, A375516, A375517.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*a(n))) end:
a:= proc(n) a(n):= 1+floor(1/((1-s(n-1))*n)) end:
seq(a(n), n=1..11);  # Alois P. Heinz, Oct 18 2024

s[n_] := s[n] = If[n == 0, 0, s[n - 1] + 1/(n*a[n])];
a[n_] := 1 + Floor[1/((1 - s[n - 1])*n)];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

{ t = 0; for (n = 1, 11, for (v = ceil(1/(n*(1-t))), oo, if (t + 1/(n*v) < 1, t += 1/(n*v); print1 (v", "); break;););); }

from itertools import count, islice
from math import gcd
def A374663_gen(): # generator of terms
    p, q = 0, 1
    for k in count(1):
        yield (m:=q//(k*(q-p))+1)
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
A374663_list = list(islice(A374663_gen(),11)) # Chai Wah Wu, Aug 28 2024

The ratios a(n)^2/a(n+1) are very close to the values 2, 2, 1, 8/5, 1/2, 7/6, 48/49, 9/8, 10/9, 11/10, 24/11^2, 13/12, 56/13^2, ... So it seems that often (but not always), a(n+1) is very close to (n/(n+1))*a(n)^2. - N. J. A. Sloane, Sep 08 2024

A375531 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} k!/a(k) < 1.

Original entry on oeis.org

2, 5, 61, 14641, 1071721201, 6891517989606967201, 332451141407535184183280941400379650401, 884190091385383640998709844252171404846723555306050253676905585566612798483201

Offset: 1

N. J. A. Sloane, Sep 04 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..11
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. A000142, A374663, A374983/A375516, A375532.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+n!/a(n)) end:
a:= proc(n) a(n):= 1+floor(n!/(1-s(n-1))) end:
seq(a(n), n=1..8);  # Alois P. Heinz, Oct 18 2024

s[n_] := s[n] = If[n == 0, 0, s[n - 1] + n!/a[n]];
a[n_] := a[n] = 1 + Floor[n!/(1 - s[n - 1])];
Table[a[n], {n, 1, 8}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)

B(u)={my(v=vector(#u)); my(r=1); for(i=1, #u, my(t=floor(u[i]/r)+1); v[i]=t; r-=u[i]/t); v}
a(n)={B(vector(n, k, k!))[n]} \\ Andrew Howroyd, Sep 04 2024

a(n) = n!*A375532(n-1) + 1.

A375532 a(n) is the denominator of Sum_{k = 1..n} k! / A375531(k).

Original entry on oeis.org

1, 2, 10, 610, 8931010, 9571552763343010, 65962528057050631782397012182615010, 21929317742693046651753716375301870159888977066122278116986745673775119010

Offset: 0

N. J. A. Sloane, Sep 04 2024

			The first few sums are 0/1, 1/2, 9/10, 609/610, 8931009/8931010, 9571552763343009/9571552763343010, ...

Alois P. Heinz, Table of n, a(n) for n = 0..10
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. A000142, A374663, A374983/A375516, A375531.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+n!/b(n)) end:
b:= proc(n) b(n):= 1+floor(n!/(1-s(n-1))) end:
a:= n-> denom(s(n)):
seq(a(n), n=0..7);  # Alois P. Heinz, Oct 18 2024

s[n_] := s[n] = If[n == 0, 0, s[n-1] + n!/b[n]];
b[n_] := b[n] = 1 + Floor[n!/(1 - s[n-1])];
a[n_] := Denominator[s[n]];
Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Dec 10 2024, after Alois P. Heinz *)

a(n+1) = (n+1)!*a(n)^2 + a(n), with a(1) = 2.

a(0)=1 prepended by Alois P. Heinz, Oct 18 2024

A376048 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} = 3,1,4,1,5,... are the digits of Pi (cf. A000796).

Original entry on oeis.org

4, 5, 81, 1621, 13130101, 310319170452181, 21399552788917656689963823241, 1373822578697020375503379392874191898311737749943783762521

Offset: 1

N. J. A. Sloane, Sep 13 2024

Rémy Sigrist and N. J. A. Sloane, Dampening Down a Divergent Series, Manuscript in preparation, September 2024.

Alois P. Heinz, Table of n, a(n) for n = 1..12

Cf. A000796, A375531, A375532.

See also A374663, A375516, A375781, A375522, A376048-A376062.

For Maple code for all these sequences, see A376056.

a(n+1) = b(n+1)*A376049(n) + 1.

A374983 a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A374663(k)).

Original entry on oeis.org

0, 1, 3, 11, 47, 1199, 241199, 9696481199, 11752718467440661199, 15347376819435640471203267700016821199, 23554197523775043569951631809272942045755944094320810352530343995293765199

Offset: 0

Rémy Sigrist, Aug 04 2024

For the denominators see A375516 and A375517.
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.
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.)

			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.

N. J. A. Sloane, Table of n, a(n) for n = 0..14
Rémy Sigrist, Proof of Theorem, Aug 26 2024, revised Sep 01 2024.
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. A374663, A375516 (denominators), 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-> numer(s(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 18 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_] := Numerator[s[n]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)

{ 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;););); }

from itertools import count, islice
from math import gcd
def A374983_gen(): # generator of terms
    p, q = 0, 1
    for k in count(1):
        yield p
        m = q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
A374983_list = list(islice(A374983_gen(),11)) # Chai Wah Wu, Aug 28 2024

A376062 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence {7/6, 5/4, 5/4, 5/4, ...}.

Original entry on oeis.org

2, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693

Offset: 1

N. J. A. Sloane, Sep 14 2024

nonn

This sequence and A376186 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.

This is essentially the same sequence as A004168 and A082732.

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. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376061, A376185.

Join[{2}, RecurrenceTable[{a[n+1] == a[n]^2 - a[n] + 1, a[2] == 4}, a, {n, 2, 9}]] (* Amiram Eldar, Sep 15 2024 *)

a(n+1) = a(n)^2 - a(n) + 1 for n >= 2.

cp's OEIS Frontend

A375517 a(n) = A375516(n)/n.

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A375791 a(n) = A375516(n+1) / A375516(n).

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A375518 First differences of A375516.

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

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A374663 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (k*a(k)) < 1.

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A375531 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} k!/a(k) < 1.

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A375532 a(n) is the denominator of Sum_{k = 1..n} k! / A375531(k).

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