A375781 -id:A375781 - cp's OEIS frontend

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

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

A375521 a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A375781(k)).

Original entry on oeis.org

0, 1, 5, 14, 103, 1154, 1336333, 892896284279, 398631887241408183843479, 19863422690705846097977473796903171171326157279, 14091270035344566960604487534521565339065390839583445590118556137472614250693240040301050079

Offset: 0

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

			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, A375522.

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-> 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/(Prime[n]*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*Prime[n])];
a[n_] := Numerator[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 A375521_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 p
A375521_list = list(islice(A375521_gen(),11)) # Chai Wah Wu, Aug 30 2024

a(0)=0 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.

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.

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.

A376052 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2k+1)a(k)) < 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 6, 31, 1527, 3509710, 19634198420529, 670572652324570519822017836, 444183929825540926086588009989665668909119960123355423

Offset: 1

N. J. A. Sloane, Sep 14 2024

nonn

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376051, A376053, A376054, A376055.

A376056 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} (2*k-1)/a(k) < 1.

Original entry on oeis.org

2, 7, 71, 6959, 62255215, 4736981006316791, 26518805245879857416837904442871, 811438882694890436523185183518581584358651922339197834228784351

Offset: 1

N. J. A. Sloane, Sep 14 2024

Theorem: Given any sequence of nonnegative integers b(1), b(2), b(3), ..., let a(1), a(2), a(3), ... be the lexicographically earliest sequence of positive integers such that for all n >= 1, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1. Then S(n) = (e(n)-1)/e(n) for positive integers e(1), e(2), e(3), ....

For the present sequence the e(k) are given in A376057.

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376055, A376057-A376061.

# Given a sequence b(1), b(2), b(3), ... of nonnegative real numbers, this program computes the first M terms of the lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... with the property that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1.
# For the present sequence we set b(k) = 2*k - 1.
b := Array(0..100,-1); a := Array(0..100,-1); S := Array(0..100,-1); d := Array(0..100,-1);
for k from 1 to 100 do b[k]:=2*k-1; od:
M:=8;
S[0] := 0; d[0] := 1;
for n from 1 to M do
a[n] := floor(b[n]/d[n-1])+1;
S[n] := S[n-1] + b[n]/a[n];
d[n] := 1 - S[n];
od:
La:=[seq(a[n],n=1..M)]; # the present sequence
Ls:=[seq(S[n],n=1..M)]; # the sums S(n)
Lsn:=[seq(numer(S[n]),n=1..M)];
Lsd:=[seq(denom(S[n]),n=1..M)]; # A376057
Lsd-Lsn; # As a check, by the above theorem, this should (and does) produce the all-1's sequence
# Some small changes to the program are needed if the starting sequence {b(n)} has offset 0, as for example in the case of the Fibonacci or Catalan numbers (see A376058-A376061).

a(n+1) = (2*n+1)*A376057(n) + 1.

A376057 a(n) is the denominator of the sum S(n) defined in A376056.

Original entry on oeis.org

1, 2, 14, 994, 6917246, 430634636937890, 2039908095836912108987531110990, 54095925512992695768212345567905438957243461489279855615252290

Offset: 0

N. J. A. Sloane, Sep 14 2024

			The first few values of S(n) are 0/1, 1/2, 13/14, 993/994, 6917245/6917246, 430634636937889/430634636937890, ...

Alois P. Heinz, Table of n, a(n) for n = 0..11

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376056.

a:= proc(n) a(n):= `if`(n=0, 1, ((2*n-1)*a(n-1)+1)*a(n-1)) end:
seq(a(n), n=0..7);  # Alois P. Heinz, Oct 18 2024

RecurrenceTable[{a[n+1] == (2*n+1)*a[n]^2 + a[n], a[0] == 1}, a, {n, 0, 7}] (* Amiram Eldar, Sep 15 2024 *)

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

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

A376058 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} Fibonacci(k)/a(k) < 1.

Original entry on oeis.org

1, 2, 3, 13, 235, 91651, 13439702641, 293516611480726842391, 139168617347514378219313352146196398680331, 31357558945249615124049146384908197437748687514518843725326663348294514909787525421

Offset: 0

N. J. A. Sloane, Sep 14 2024

Cf. A000045, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376057, A376059

a(n+1) = Fibonacci(n+1)*A376059(n) + 1.

A376061 a(n) is the denominator of the sum S(n) defined in A376060.

Original entry on oeis.org

2, 6, 78, 30498, 13021822554, 7121850230383271305026, 6695139092929353602428277531338786356808914258

Offset: 0

N. J. A. Sloane, Sep 14 2024

			The first few values of S(n) are 1/2, 5/6, 77/78, 30497/30498, 13021822553/13021822554, ...

Cf. A000108, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376060.

RecurrenceTable[{a[n+1] == CatalanNumber[n+1]*a[n]^2 + a[n], a[0] == 2}, a, {n, 0, 6}] (* Amiram Eldar, Sep 15 2024 *)

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

cp's OEIS Frontend

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

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A375521 a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A375781(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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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, ...}.

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

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A376052 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2*k+1)*a(k)) < 1.

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A376056 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} (2*k-1)/a(k) < 1.

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A376057 a(n) is the denominator of the sum S(n) defined in A376056.

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A376058 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} Fibonacci(k)/a(k) < 1.

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A376052 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2k+1)a(k)) < 1.