A374663 -id:A374663 - cp's OEIS frontend

A376244 Lexicographically earliest sequence of positive integers a(1), a(2), ... with the property that the lexicographically earliest sequence of positive integers b(1), b(2), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1 / (a(k)*b(k)) < 1, also implies that S(n) is never of the form (e_n - 1) / e_n for some integer e_n.

3, 4, 5, 4, 7, 3, 9, 1, 11, 4, 13, 7, 9, 19, 10, 2, 23, 25, 29, 27, 53, 1, 17, 7, 2, 2, 15, 67, 22, 37

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Sep 16 2024

Is this sequence infinite?

			The initial terms are:
  n  a(n)  b(n)    S(n)
  -  ----  ------  ---------------------------
  1     3       1  1/3
  2     4       1  7/12
  3     5       1  47/60
  4     4       2  109/120
  5     7       2  823/840
  6     3      17  4757/4760
  7     9     177  7582661/7582680
  8     1  399089  3026164178509/3026164178520

Rémy Sigrist, PARI program

Cf. A374663, A376062, A376184, A376245 (corresponding b's), A376246-A376247 (numerator and denominator of corresponding S(n)).

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A375834 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(k*a(n+1-k)) < 1.

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6

Offset: 1

Rémy Sigrist, Aug 31 2024

nonn

This sequence is unbounded.

			The first terms, alongside the corresponding sums, are:
  n   a(n)  Sum {k=1..n} 1/(k*a(n+1-k))
  --  ----  ---------------------------
   1     2  1/2
   2     2  3/4
   3     2  11/12
   4     3  7/8
   5     3  107/120
   6     3  331/360
   7     3  299/315
   8     3  4931/5040
   9     4  4651/5040
  10     3  4993/5040
  11     4  26219/27720
  12     4  155389/166320
  13     4  201613/216216
  14     4  288793/308880
  15     4  2031847/2162160

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A374663, A375814, A375815.

{ for (n = 1, #a = vector(87), a[n] = floor(1/(1-sum(k = 2, n, 1/(k*a[n+1-k])))) + 1; print1 (a[n]", ");); }

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.

A377205 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(k^2*a(k)) < 1.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 3, 9, 171, 122014, 17661589931, 412924014578486602517, 1248808068140660770289141544749321839183623, 4529027355107615424925871833487047912228337079416162414871862143803627237910792872226

Offset: 1

Alois P. Heinz, Oct 19 2024

nonn

			s(0), s(1), ... = 0, 1/2, 3/4, 31/36, 133/144, 3469/3600, 3569/3600, ... .

Alois P. Heinz, Table of n, a(n) for n = 1..17

Cf. A000290, A013661, A374663, A377229, A377230.

s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+1/(n^2*a(n))) end:
a:= proc(n) option remember; 1+floor(1/((1-s(n-1))*n^2)) end:
seq(a(n), n=1..14);

A377229 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(F(k)*a(k)) < 1, F = Fibonacci.

Original entry on oeis.org

2, 3, 4, 9, 44, 1486, 1357976, 1855074754595, 2975714380792664939835466, 46528348836004781630107949818181021469921360198769

Offset: 1

Alois P. Heinz, Oct 20 2024

nonn

			s(0), s(1), ... = 0, 1/2, 5/6, 23/24, 215/216, 11879/11880, 17653679/17653680, ... .

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

Cf. A000045, A079586, A374663, A376058, A377205, A377230.

F:= combinat[fibonacci]:
s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+1/(F(n)*a(n))) end:
a:= proc(n) option remember; 1+floor(1/((1-s(n-1))*F(n))) end:
seq(a(n), n=1..11);

A377230 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(T(k)*a(k)) < 1, T = A000217.

Original entry on oeis.org

2, 1, 2, 2, 3, 5, 23, 806, 519065, 220441054222, 222723684271305542570701, 41974171914555858099300698444579076459265512901, 1510140949639448391630842209382251970116940997822995817347241840058937174456186756365141648201

Offset: 1

Alois P. Heinz, Oct 20 2024

nonn

			s(0), s(1), ... = 0, 1/2, 5/6, 11/12, 29/30, 89/90, 629/630, ... .

Alois P. Heinz, Table of n, a(n) for n = 1..16

Cf. A000217, A374663, A377205, A377229.

T:= n-> n*(n+1)/2:
s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+1/(T(n)*a(n))) end:
a:= proc(n) option remember; 1+floor(1/((1-s(n-1))*T(n))) end:
seq(a(n), n=1..13);

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A375834 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(k*a(n+1-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/((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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A377205 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(k^2*a(k)) < 1.

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A377229 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(F(k)*a(k)) < 1, F = Fibonacci.

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Maple

A377230 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(T(k)*a(k)) < 1, T = A000217.

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.