A082732 -id:A082732 - cp's OEIS frontend

A376184 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 b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).

2, 5, 17, 341, 92753, 10753782821, 92515075960384748177, 10698799099944699918936107506299150093941, 91571441744782016867976366392607084634231243149599342901251284090792487979854033

Offset: 1

N. J. A. Sloane, Sep 15 2024

This sequence and A376062 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.

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...

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-A376062, A376186.

A376186 a(n) = denominator of the sum S(n) defined in A376184.

Original entry on oeis.org

8, 40, 680, 231880, 21507565640, 231287689900961870440, 21397598199889399837872215012598300187880, 228928604361955042169940915981517711585578107873998357253128210226981219949635080

Offset: 1

N. J. A. Sloane, Sep 15 2024

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...

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-A376062, A376184, A376185.

A275698 a(0) = 2, after that a(n) is 3 plus the least common multiple of previous terms.

Original entry on oeis.org

2, 5, 13, 133, 17293, 298995973, 89398590973228813, 7992108067998667938125889533702533, 63873791370569400659097694858350356285036046451665934814399129508493

Offset: 0

Andres Cicuttin, Aug 05 2016

nonn

This sequence could be considered a particular case of a possible two-parameter family of sequences of the form: a(n) = k1 + lcm(a(0),a(1),..,a(n-1)), a(0) = k2, where in this case k1=3 and k2=2. With other choices of k1 and k2 it seems it is possible to generate other sequences such as
A129871 with k1 = 1 and k2 = 1,
A000058 with k1 = 1 and k2 = 2,
A082732 with k1 = 1 and k2 = 3,
A000215 with k1 = 2 and k2 = 3,
A000324 with k1 = 4 and k2 = 1,
A001543 with k1 = 5 and k2 = 1,
A001544 with k1 = 6 and k2 = 1,
A275664 with k1 = 2 and k2 = 2,
A000289 with k1 = 3 and k2 = 1.

S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A082732, A275664.

A275698 = {2}; Do[AppendTo[A275698, 3 + LCM@@A275698], {i, 9}]; A275698

a(n) = 3 + lcm(a(0), a(1), ..., a(n - 1)), a(0) = 2.
a(n) = 3 + a(n-1)*(a(n-1)-3), for n > 1. - Christian Krause, Oct 17 2023. Proof: Follows from associativity of lcm(...) and the fact that gcd(m,m+3)=1:
a(n)-3 = lcm(a(0),a(1),...,a(n-2),a(n-1))
       = lcm(lcm(a(0),a(1),...,a(n-2)),a(n-1))
       = lcm(a(n-1)-3,a(n-1))
       = (a(n-1)-3)*a(n-1).

A174954 a(1)=1 and a(2)=2, a(n) = square of the sum of previous terms.

Original entry on oeis.org

1, 2, 9, 144, 24336, 599858064, 359859080993093136, 129498558604939936508538275302878864

Offset: 1

Giovanni Teofilatto, Apr 02 2010

Sqrt(a(n+1)/a(n)) = A082732(n).

Cf. A082732, A174864.

cp's OEIS Frontend

A376184 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 b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).

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A376186 a(n) = denominator of the sum S(n) defined in A376184.

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A275698 a(0) = 2, after that a(n) is 3 plus the least common multiple of previous terms.

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Formula

A174954 a(1)=1 and a(2)=2, a(n) = square of the sum of previous terms.

Views

Author

Keywords

Comments

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cp's OEIS Frontend

A376184 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 b(1)=5/4, b(2*i)=3/2, b(2*i+1)=6/5 (i>0).

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Author

Keywords

Comments

Examples

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Crossrefs

A376186 a(n) = denominator of the sum S(n) defined in A376184.

Views

Author

Keywords

Examples

Links

Crossrefs

A275698 a(0) = 2, after that a(n) is 3 plus the least common multiple of previous terms.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A174954 a(1)=1 and a(2)=2, a(n) = square of the sum of previous terms.

Views

Author

Keywords

Comments

Crossrefs

A376184 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 b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).