A374983 -id:A374983 - cp's OEIS frontend

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.

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

A375516 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A374663(k)).

Original entry on oeis.org

1, 2, 4, 12, 48, 1200, 241200, 9696481200, 11752718467440661200, 15347376819435640471203267700016821200, 23554197523775043569951631809272942045755944094320810352530343995293765200

Offset: 0

N. J. A. Sloane, Aug 19 2024

In fact a(n) = A374983(n) + 1 (see the proof in A374983), but this was unproved when this sequence was created, and in any case the prime factors of A374983(n) and a(n) are both of interest, so both sequences are included in the OEIS. Both sequences grow doubly exponentially. See also A375791.

One might be led to conjecture that the last 4 digits of the numbers from a(5) onwards are always 1200, but Rémy Sigrist has observed that this does not hold for a(10) = 23554197523775043569951631809272942045755944094320810352530343995293765200.

N. J. A. Sloane, Table of n, a(n) for n = 0..14
Rémy Sigrist, Proof of theorem about A374983 and the present sequence, 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, A374983.

See A375517 for a(n)/n and A375791 for a(n+1)/a(n).

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)):
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_] := Denominator[s[n]];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Dec 10 2024, after Alois P. Heinz *)

from itertools import count, islice
from math import gcd
def A375516_gen(): # generator of terms
    p, q = 0, 1
    for k in count(1):
        yield q
        m = q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
A375516_list = list(islice(A375516_gen(),11)) # Chai Wah Wu, Aug 28 2024

A375522 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A375781(k)).

Original entry on oeis.org

1, 2, 6, 15, 105, 1155, 1336335, 892896284280, 398631887241408183843480, 19863422690705846097977473796903171171326157280, 14091270035344566960604487534521565339065390839583445590118556137472614250693240040301050080

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Aug 30 2024

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.
The differences S2(n) - S1(n) are surprisingly small: for n = 1,2,...,34 the values S2(n) - S1(n) are:
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
suggesting the conjecture that they are always 1 except for n = 4 and 6 (compare the Theorem in A374983).

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

Alois P. Heinz, Table of n, a(n) for n = 0..13
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. A375781, A375521.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(ithprime(n)*b(n))) end:
b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*ithprime(n))) end:
a:= n-> denom(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/(Prime[n]*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*Prime[n])];
a[n_] := Denominator[s[n]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)

from itertools import islice
from math import gcd
from sympy import nextprime
def A375522_gen(): # generator of terms
    p, q, k = 0, 1, 1
    while (k:=nextprime(k)):
        m=q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
        yield q
A375522_list = list(islice(A375522_gen(),11)) # Chai Wah Wu, Aug 30 2024

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

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

Original entry on oeis.org

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

A375529 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} prime(k)/a(k) < 1.

Original entry on oeis.org

3, 10, 151, 31711, 1580159131, 2950885219102973491, 11387023138265143513338462726052139311, 144918919004489964473283047921945994420315076260338720025368711042369934871

Offset: 1

Rémy Sigrist and 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. A374663, A374983/A375516, A375530.

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,prime(k)))[n]} \\ Andrew Howroyd, Sep 04 2024

a(n) = prime(n)*A375530(n-1) + 1.

A375530 a(n) is the denominator of Sum_{k = 1..n} prime(k) / A375529(k).

Original entry on oeis.org

1, 3, 30, 4530, 143650830, 226991170700228730, 669824890486184912549321336826596430, 7627311526552103393330686732733999706332372434754669475019405844335259730

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Sep 04 2024

			The first few fractions Sum_{k = 1..n} prime(k) / A375529(k) are 0/1, 2/3, 29/30, 4529/4530, 143650829/143650830, 226991170700228729/226991170700228730, ...

Alois P. Heinz, Table of n, a(n) for n = 0..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. A374663, A374983/A375516, A375529.

a:= proc(n) option remember; `if`(n=0, 1,
      ithprime(n)*a(n-1)^2+a(n-1))
    end:
seq(a(n), n=0..7);  # Alois P. Heinz, Oct 21 2024

a(n) = prime(n)*a(n-1)^2 + a(n-1), with a(0) = 1.

a(0)=1 prepended by Alois P. Heinz, Oct 21 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

cp's OEIS Frontend

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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A375516 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A374663(k)).

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A375522 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A375781(k)).

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

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A375529 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} prime(k)/a(k) < 1.

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