A376048 -id:A376048 - cp's OEIS frontend

A376049 a(n) is the denominator of the sum S(n) defined in A376048.

1, 4, 20, 1620, 2626020, 34479907828020, 10699776394458828344981911620, 228970429782836729250563232145698649718622958323963960420, 314564746289621790478499809054644383073494124885364231904506529162346525698245870188461228508418377184391923418820

Offset: 0

N. J. A. Sloane, Sep 13 2024

			The first few values of S(n) are 0/1, 3/4, 19/20, 1619/1620, 2626019/2626020, 34479907828019/34479907828020, ...

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

Cf. A000796, A375531, A375532, A376048.

a(n+1) = b(n+1)*a(n)^2 + a(n), with a(1) = 4, where b(n) is defined in A376048.

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

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.

A376050 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

2, 1, 2, 3, 6, 172, 137534, 106557767317, 10018727448950607892211, 218107864753736742334588510315735629277159621, 43040465365773907074907163986022284668974202910116417170603263409796800986397420975160781

Offset: 1

N. J. A. Sloane, Sep 13 2024

It appears that S(n) = (e(n)-1)/e(n) for all n != 4, where e(n) = A376051(n). Exceptionally, S(4) = (e(4)-2)/e(4).

a(15) has 1420 decimal digits, too large for a b-file. - Robert Israel, Oct 13 2024

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

Robert Israel, Table of n, a(n) for n = 1..14

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048, A376049, A376051.

S:= 1:R:= NULL:
for i from 1 to 11 do
  r:= ceil(1/((2*i-1)*S));
  if r *(2*i-1) = 1/S then r:= r+1 fi;
  R:= R,r;
  S:= S - 1/((2*i-1)*r)
od:
R; # Robert Israel, Oct 13 2024

A376053 Numerator of the sum S(n) defined in A376052.

Original entry on oeis.org

1, 8, 71, 248, 3043, 43024, 89051, 764441, 451021514, 25508567769, 411827311870583771, 525058386770138717020639964821, 528134692562568161116953143877712480332943632586669596859, 2267693117789905604207315326366543773113615946806750184592188584359364943382168221068055512231683584106110223751

Offset: 1

N. J. A. Sloane, Sep 14 2024

			The initial values of S(n) are 1/3, 8/15, 71/105, 248/315, 3043/3465, 43024/45045, 89051/90090, ...

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376051, A376052, A376054, A376055

cp's OEIS Frontend

A376049 a(n) is the denominator of the sum S(n) defined in A376048.

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

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A376050 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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Maple

A376053 Numerator of the sum S(n) defined in A376052.

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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.

A376050 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.