A375516 -id:A375516 - cp's OEIS frontend

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.

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.

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.

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

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

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

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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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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.