A376049 -id:A376049 - cp's OEIS frontend

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).

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.

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.

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

cp's OEIS Frontend

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

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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/((2k-1)a(k)) < 1.

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cp's OEIS Frontend

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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Author

Keywords

References

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Examples

Crossrefs

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

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.