A226049 -id:A226049

A226052

Denominators of signed Egyptian fractions with sums converging to sqrt(2).

+10
4

2, 11, 195, 180120, 120479425978, 27716921130006533867139, 1040296455490146050257045342043017466273633682

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Using the algorithm defined at A226049 with r = sqrt(2) and f(n) = 1/n gives r = 1 + 1/2 - 1/11 + 1/195 - 1/180120 + ..., of which the 13th partial sum differs from the r by less than 10^(-2900). For a guide to related sequences, see A226049.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..12

EXAMPLE

Let r = sqrt(2). Then

1 < r < 1 + 1/2, so a(1) = 2.

1 + 1/2 -1/11 < r, so a(2) = 11.

1 + 1/2 - 1/11 + 1/195 > r, so a(3) = 195.

MATHEMATICA

$MaxExtraPrecision = Infinity;

nn = 12; f[n_] := 1/n; r = Pi; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 24 2013

STATUS

approved

A226050

Denominators of signed Egyptian fractions with sums converging to e.

+10
2

9, 9, 2354, 8114635, 95238369598066, 10137142998831712366552473861, 299020940111770751476683910911849997367708901463842358682

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Using the algorithm defined at A226049 with r = e and f(n) = 1/n gives

r = sum{1/k, k=1..9} - 1/9 + 1/2354 - 1/8114635 + ...

r = sum{1/k, k=1..8} + 1/2354 - 1/81154635 + ...; for this second series, the 17th partial sum differs from the e by less than 10^(-900). For a guide to related sequences, see A226049.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..12

EXAMPLE

1 + 1/2 + ... + 1/8 < e < 1 + 1/2 + ... + 1/8 + 1/9, so a(1) = 9.

1 + 1/2 + ... + 1/9 - 1/9 < e, so a(2) = 9.

1 + 1/2 + ... + 1/9 - 1/9 + 1/2354 > e, so a(3) = 2354.

MATHEMATICA

$MaxExtraPrecision = Infinity;

nn = 10; f[n_] := 1/n; r = E; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 24 2013

STATUS

approved

A226051

Denominators of signed Egyptian fractions with sums converging to Pi.

+10
2

13, 25, 685, 1046508, 3249663242463, 11242879629599233747822858, 296966610697839275946742496484126789076162611540968

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Using the algorithm defined at A226049 with r = Pi and f(n) = 1/n gives r = (Sum_{k=1..13} 1/k) - 1/25 + 1/685 - 1/1046508 + ..., of which the 22nd partial sum differs from Pi by less than 10^(-800). For a guide to related sequences, see A226049.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..12

EXAMPLE

1 + 1/2 + ... + 1/12 < Pi < 1 + 1/2 + ... + 1/13, so a(1) = 13.

1 + 1/2 + ... + 1/13 - 1/25 < Pi, so a(2) = 25.

1 + 1/2 + ... + 1/13 - 1/25 + 1/685 > Pi, so a(3) = 685.

MATHEMATICA

$MaxExtraPrecision = Infinity;

nn = 10; f[n_] := 1/n; r = Pi; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 24 2013

STATUS

approved

A226053

Denominators of signed Egyptian fractions with sums converging to sqrt(1/2).

+10
2

1, 3, 24, 815, 2263886, 9073564639850, 176228569027146222763928594, 84205747605016031994416006285857418872429042805656089

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Using the algorithm defined at A226049 with r = sqrt(1/2) and f(n) = 1/n gives r = 1 + 1/2 + 1/3 - 1/24 + 1/815 + 1/2263886 + ..., of which the 14th partial sum differs from the r by less than 10^(-1700). For a guide to related sequences, see A226049.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..12

EXAMPLE

Let r = sqrt(1/2). Then

1 + 1/2 < r < 1 + 1/2 + 1/3, so a(1) = 3.

1 + 1/2 + 1/3 -1/24 < r, so a(2) = 24.

1 + 1/2 + 1/3 -1/24 + 1/815 > r, so a(3) = 815.

MATHEMATICA

$MaxExtraPrecision = Infinity;

nn = 12; f[n_] := 1/n; r = Sqrt[1/2]; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 24 2013

STATUS

approved

A226058

Denominators of signed Egyptian fractions with sums converging to the Euler-Mascheroni constant.

+10
2

1, 2, 12, 163, 57800, 6910401969, 69047272940549306735, 9431570193753590001558014495854550084583

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Using the algorithm defined at A226049 with r = Euler-Mascheroni constant (0.577216...) and f(n) gives r = 1 - 1/2 + 1/12 - 1/163 + 1/57800 + ..., of which the 12th partial sum differs from the r by less than 10^(-1900). For a guide to related sequences, see A226049.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..11

EXAMPLE

Let r = Euler-Mascheroni constant. Then

r < 1 , so a(1) = 1.

1 - 1/2 < r, so a(2) = 2.

1 - 1/2 + 1/12 > r, so a(3) = 12.

MATHEMATICA

$MaxExtraPrecision = Infinity;

nn = 12; f[n_] := 1/n; r = EulerGamma; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 24 2013

STATUS

approved

A226124

Denominators of signed reciprocal primes with sums converging to 1.

+10
1

2, 3, 5, 29, 863, 107251, 1519341947, 1028110904472525023, 18408375793288791212165960272609427, 12833942597163525070720542528785599268668682079686196789779156965969

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The algorithm at A226049, with r = 1 and f(n) = 1/prime(n), gives the sum 1/2 + 1/3 + 1/5 - 1/29 + 1/863 - 1/107251 + 1/1519341947 - ... = 1, of which the denominators on the left side comprise this sequence.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..14

EXAMPLE

1/2 + 1/3 + 1/5 - 1/29 + 1/863 - 1/107251 + 1/1519341947 differs from 1 by less than 10^(-18).

MATHEMATICA

p[n_] := Prime[n]; q[x_] := PrimePi[x]; r = 1; u = 1/2 + 1/3 + 1/5; c[1] = q[1/(u - r)]; c[2] = q[1/(r - u + 1/p[c[1]])]; c[3] = q[1/(u - r - 1/p[c[1]] + 1/p[c[2]])]; c[4] = q[1/(r - u + 1/p[c[1]] - 1/p[c[2]] + 1/p[c[3]])]; Union[{2, 3, 5}, Table[p[c[i]], {i, 1, 4}]]

seq={2, 3, 5}; sum=Total@(1/seq); While[Length[seq]<10, p=NextPrime[Abs[1/(sum-1)], -1]; sum+=Sign[1-sum]/p; AppendTo[seq, p]]; seq (* Amiram Eldar, Mar 13 2019 *)

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 27 2013

EXTENSIONS

a(8)-a(10) from Amiram Eldar, Mar 13 2019

STATUS

approved

A226125

Denominators of signed Egyptian fractions 1/(1+a(n)) with sums converging to sqrt(2).

+10
1

5, 26, 798, 1036219, 1679769509872, 6521391013068322601901238, 285305237462037970325184857571103801772580723343165

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The algorithm at A226049, with r = sqrt(2), f(n) = 1/(n+1), gives

1 + 1/2 + ... + 1/a(1) - 1/a(2) + 1/a(3) - , ... converging to sqrt(2). The sum 1 + 1/2 + ... - 1/a(12) differs from sqrt(2) by less than 10^(-3200).

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

Sum of the first 7 signed Egyptian fractions: 1/2 + 1/3 + 1/5 + 1/6 - 1/27 + 1/799 - 1/1036229, showing denominators (beginning at 6), a(1)+1, a(2)+1, a(3)+1, ...

MATHEMATICA

$MaxExtraPrecision = Infinity; z = 12; f[n_] := 1/(n + 1); g[n_] := (1 - n)/n; r = Sqrt[2]; s = 0; a[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; p = Sum[f[n], {n, 1, a[1]}]; a[2] = Floor[g[p - r]]; a[n_] :=

Floor[g[((-1)^n) (p - r - Sum[((-1)^k) f[a[k]], {k, 2, n - 1}])]]; Table[a[k], {k, 1, z}]

CROSSREFS

Cf. A226049, A226052, A226126.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 27 2013

STATUS

approved

A226126

Denominators of signed Egyptian fractions 1/(2+a(n)) with sums converging to sqrt(2).

+10
1

8, 65, 5856, 39703530, 1895307350287177, 46342142299686404785297514402543, 186913643198872746939347285190966650469716954961536584709363654945

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The algorithm at A226049, with r = sqrt(2) and f(n) = 1/(n+2), gives a sum that converges to sqrt(2). The 16th partial sum differs from sqrt(2) by less than 10^(-500).

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

Sum of the first 12 signed Egyptian fractions: 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 - 1/67 + 1/5858 - 1/39703532 showing denominators (beginning at 10), a(1)+1, a(2)+1, a(3)+1, ...

MATHEMATICA

$MaxExtraPrecision = Infinity;

z = 9; f[n_] := 1/(n + 2); g[n_] := 1/n - 2; r = Sqrt[2]; s = 0; a[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; p = Sum[f[n], {n, 1, a[1]}]; a[2] = Floor[g[p - r]]; a[n_] := Floor[g[((-1)^n) (p - r - Sum[((-1)^k) f[a[k]], {k, 2, n - 1}])]]; Table[a[k], {k, 1, z}]

N[p - Sum[((-1)^n)*f[a[n]], {n, 2, z}] - r, 20]

CROSSREFS

Cf. A226049, A226052, A226125.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 27 2013

STATUS

approved

A226128

Denominators of signed Egyptian fractions 1/(1+2*a(n)) with sums converging to 2.

+10
1

56, 151, 37675, 162957309354, 96984274430119214824218, 10080078376423662538051091122673282619968956943

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A226049.

LINKS

Table of n, a(n) for n=1..6.

EXAMPLE

The algorithm at A226049, with r = 2 and f(n) = 1/(2n+1), gives a sum

1/3 + 1/5 + 1/7 + ... + 1/(1+2*56) - 1/(1+2*151) + 1/(1+2*37675) - ...

that converges to 2. The 64th partial sum differs from 2 by less than 10^(-700).

MATHEMATICA

$MaxExtraPrecision = Infinity; z = 9; f[n_] := 1/(2 n + 1); g[n_] := (1/n - 1)/2; r = 2; s = 0; a[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; p = Sum[f[n], {n, 1, a[1]}]; a[2] = Floor[g[p - r]]; a[n_] := Floor[g[((-1)^n) (p - r - Sum[((-1)^k) f[a[k]], {k, 2, n - 1}])]];

Table[a[k], {k, 1, z}]

CROSSREFS

Cf. A226049, A226128.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 27 2013

STATUS

approved

A226127

Denominators of signed Egyptian fractions 1/(1+2*a(n)) with sums converging to sqrt(2).

+10
0

17, 38, 2558, 53508058, 10183965708276283, 833167602683818992272386593114136, 7008824222617646742567474710166940582408242437487752499090108838014

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A226049.

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

The algorithm at A226049, with r = sqrt(2), f(n) = 1/(2n+1), gives

1/3 + 1/5 + 1/7 + ... + 1/(1+2*17) - 1/(1+2*38) + 1/(1+2*2558) - ...

which converges to sqrt(2). The 17th partial sum differs from sqrt(2) by less than 10^(-500).

MATHEMATICA

$MaxExtraPrecision = Infinity; z = 9; f[n_] := 1/(2 n + 1); g[n_] := (1/n - 1)/2; r =

Sqrt[2]; s = 0; a[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; p = Sum[f[n], {n, 1, a[1]}]; a[2] = Floor[g[p - r]]; a[n_] := Floor[g[((-1)^n) (p - r - Sum[((-1)^k) f[a[k]], {k, 2, n - 1}])]]; Table[a[k], {k, 1, z}]

CROSSREFS

Cf. A226049.

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 27 2013

STATUS

approved