A374663 -id:A374663 - cp's OEIS frontend

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

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

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

A375781 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (prime(k)*a(k)) < 1 (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 1, 2, 3, 5, 89, 39304, 46994541278, 17331821184409051471456, 684945610024339520619912889548385212804350252, 454557097914340869696918952726502107711786801276885341616727617337826266151394840009711293

Offset: 1

Rémy Sigrist, Aug 28 2024

nonn

The sum of the reciprocals of the primes diverges. We divide each of its terms in such a way as to have a series bounded by 1.

			The first terms, alongside the corresponding sums, are:
  n  a(n)   Sum_{k=1..n} 1/(prime(k)*a(k))
  -  -----  ------------------------------
  1      1  1/2
  2      1  5/6
  3      2  14/15
  4      3  103/105
  5      5  1154/1155
  6     89  1336333/1336335
  7  39304  892896284279/892896284280

Alois P. Heinz, 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]
Wikipedia, Divergence of the sum of the reciprocals of the primes

Cf. A000040, A374663.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(ithprime(n)*a(n))) end:
a:= proc(n) a(n):= 1+floor(1/((1-s(n-1))*ithprime(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/(Prime[n]*a[n])];
a[n_] := a[n] = 1 + Floor[1/((1 - s[n-1])*Prime[n])];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Mar 19 2025, after Alois P. Heinz *)

{ r = 1; forprime (p = 2, prime(11), print1 (a = floor(1/(r*p)) + 1", "); r -= 1 / (a*p);); }

from itertools import islice
from math import gcd
from sympy import nextprime
def A375781_gen(): # generator of terms
    p, q, k = 0, 1, 1
    while (k:=nextprime(k)):
        yield (m:=q//(k*(q-p))+1)
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
A375781_list = list(islice(A375781_gen(),11)) # Chai Wah Wu, Aug 30 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.

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

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.

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.

A376051 a(n) is the denominator of the sum S(n) defined in A376050.

Original entry on oeis.org

2, 6, 15, 105, 1890, 1787940, 1598366509740, 170318366632160334167580, 4144049430320998104357181695998976956266032780, 903849772681252048573050443706467978048458261112444760582668531605732820714345840478376380

Offset: 1

N. J. A. Sloane, Sep 13 2024

			The first few values of S(n) are 1/2, 5/6, 14/15, 103/105, 1889/1890, 1787939/1787940, 1598366509739/1598366509740, ... Note S(4) is exceptional, in that the numerator and denominator differ by 2 instead of 1.

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048, A376049, A376050.

cp's OEIS Frontend

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

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

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A375781 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (prime(k)*a(k)) < 1 (where prime(k) denotes the k-th prime number).

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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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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).

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

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