A013631 -id:A013631

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A033167

Positions of the incrementally largest terms in the continued fraction expansion of zeta(3), offset 1 variant.

+10
1

1, 2, 4, 29, 63, 572, 1556, 2013, 2530, 2760, 3019, 4159, 4741, 6820, 10565, 11666, 32859, 139893, 392130, 707970, 1049722, 2081165, 14990404, 36112276, 39552835, 42710787, 199618806 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`Positions in this sequence correspond to the n-th term of A013631 at index n-1.` `See A229055 for another version.`
	LINKS	`Table of n, a(n) for n=1..27.` `Eric Weisstein's World of Mathematics, Apery's Constant Continued Fraction`
	CROSSREFS	`Cf. A229055 (= a(n) - 1), A013631 (continued fraction of zeta(3)), A033165, A000023, A033166.`
	KEYWORD	`nonn,more`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`More terms from Eric W. Weisstein, Aug 23, 2000` `More terms from Robert Gerbicz, Aug 22 2006` `Edited (with more terms taken from A229055) by N. J. A. Sloane, Jun 16 2021` `Edited for offset change in A013631. - Andrew Howroyd, Jul 10 2024`
	STATUS	`approved`

A013687

Continued fraction for zeta(11).

+10
0

1, 2023, 1, 1, 12, 1, 2, 2, 1, 102, 1, 44, 1, 2, 2, 1, 2, 3, 1, 5, 2, 1, 1, 2, 1, 13, 4, 14, 2, 5, 1, 5, 1, 6, 1, 2, 9, 1, 1, 1, 1, 7, 1, 2, 3, 1, 39, 3, 119, 12, 1, 1, 5, 1, 1, 151, 3, 4, 1, 2, 4, 98, 29, 6, 2, 1, 3, 9, 1, 1, 1, 5, 1, 2 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..73.` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[11], 80] (* Harvey P. Dale, May 22 2013 *)`
	CROSSREFS	`Cf. A013669.` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 08 2024`
	STATUS	`approved`

A013689

Continued fraction for zeta(13).

+10
0

1, 8149, 13, 1, 2, 1, 6, 23, 3, 1, 7, 1, 1, 5, 1, 1, 4, 1, 1, 1, 4, 1, 1, 2, 2, 8, 1, 29, 32, 22, 2, 123, 1, 2, 1, 10, 1, 2, 2, 1, 4, 1, 13, 5, 8, 34, 2, 1, 7, 1, 2, 1, 3, 20, 8, 1, 4, 1, 5, 1, 59, 3, 7, 1, 1, 3, 2, 6, 1, 1, 2, 9, 1, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..73.` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[13], 100] (* Harvey P. Dale, Feb 25 2015 *)`
	CROSSREFS	`Cf. A013671.` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 08 2024`
	STATUS	`approved`

A013690

Continued fraction for zeta(14).

+10
0

1, 16327, 36, 19, 2, 1, 35, 1, 4, 7, 5, 1, 1, 1, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 7, 1, 4, 7, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 4, 9, 2, 2, 1, 23, 6, 1, 2, 1, 2, 1, 1, 10, 1, 19, 7, 1, 1, 42, 1, 15, 1, 1, 4, 1, 2, 2, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..75.` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[14], 80] (* Harvey P. Dale, Jun 28 2014 *)`
	CROSSREFS	`Cf. A013672.` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 08 2024`
	STATUS	`approved`

A013691

Continued fraction for zeta(15).

+10
0

1, 32692, 3, 3, 1, 4, 1, 2, 3, 2, 1, 1, 1, 1, 1, 3, 1, 5, 1, 4, 1, 54, 1, 5, 5, 1, 20, 57, 5, 8, 1, 2, 26, 1, 1, 1, 1, 10, 1, 12, 1, 1, 7, 1, 2, 4, 1, 4, 1, 3, 5, 1, 1, 1, 1, 2, 4, 1, 18, 2, 2, 4, 1, 7, 4, 5, 1, 4, 2, 1, 1, 3, 1, 5, 1, 28 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..75.` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[15], 80] (* Harvey P. Dale, Jun 01 2012 *)`
	CROSSREFS	`Cf. A013673.` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 08 2024`
	STATUS	`approved`

A013692

Continued fraction for zeta(16).

+10
0

1, 65435, 2, 1, 5, 1, 4, 1, 3, 3, 1, 7, 1, 2, 6, 2, 1, 7, 1, 1, 2, 1, 4, 4, 2, 3, 13, 1, 2, 1, 5, 1, 1, 8, 1, 5, 1, 1, 1, 4, 1, 2, 2, 2, 1, 44, 1, 2, 1, 2, 4, 2, 1, 6, 153, 41, 1, 26, 1, 4, 1, 3, 3, 1, 1, 1, 5, 6, 15, 4, 7, 1, 1, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..75.` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[16], 80] (* Harvey P. Dale, Mar 21 2012 *)`
	CROSSREFS	`Cf. A013674.` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 08 2024`
	STATUS	`approved`

A013693

Continued fraction for zeta(17).

+10
0

1, 130938, 12, 2, 2, 8, 1, 6, 2, 3, 4, 2, 6, 1, 1, 7, 3, 10, 1, 5, 1, 2, 1, 2, 33, 3, 1, 4, 1, 1, 7, 5, 7, 1, 4, 1, 6, 1, 1, 2, 1, 1, 1, 5, 1, 1, 4, 1, 1, 1, 3, 1, 1, 3, 8, 2, 2, 2, 5, 5, 4, 2, 7, 2, 45, 5, 6, 2, 1, 1, 53, 1, 1, 1, 4, 1, 2 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..76.` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[17], 100] (* Harvey P. Dale, Oct 26 2015 *)`
	CROSSREFS	`Cf. A013675.` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 08 2024`
	STATUS	`approved`

A265824

Continued fraction expansion of the prime zeta function at 3.

+10
0

0, 5, 1, 2, 1, 1, 2, 17, 4, 1, 7, 1, 1, 5, 24, 1, 1, 2, 3, 11, 1, 3, 23, 1, 1, 2, 1, 3, 1, 6, 1, 4, 3, 3, 1, 2, 1, 4, 1, 1, 3, 1, 1, 1, 2, 23, 2, 6, 2, 2, 1, 1, 7, 3, 13, 1, 1, 2, 6, 1, 5, 5, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 28, 2, 1, 4, 10, 3, 2, 1, 1, 2, 1, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	COMMENTS	`Continued fraction of Sum_{n>=1} 1/prime(n)^3 = 0.1747626392994435364231...`
	LINKS	`Table of n, a(n) for n=0..85.` `Eric Weisstein's World of Mathematics, Prime Zeta Function` `Wikipedia, Prime Zeta Function` `Index entries for continued fractions for constants` `Index entries for zeta function`
	EXAMPLE	`1/2^3 + 1/3^3 + 1/5^3 +1/7^3 + 1/11^3 + 1/13^3 +... = 1/(5 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(2 + 1/(17 + 1/(4 + 1/…)))))))).`
	MATHEMATICA	`ContinuedFraction[PrimeZetaP[3], 85]`
	CROSSREFS	`Cf. A085541, A013631.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`Ilya Gutkovskiy, Dec 16 2015`
	STATUS	`approved`

A269444

Continued fraction expansion of the Dirichlet eta function at 3.

+10
0

0, 1, 9, 6, 2, 1, 1, 1, 1, 1, 1, 6, 1, 4, 1, 7, 2, 1, 1, 1, 2, 91, 32, 1, 1, 6, 23, 1, 1, 1, 1, 2, 9, 1, 2, 1, 1, 5, 1, 1, 37, 12, 1, 12, 3, 2, 87, 1, 4, 2, 2, 2, 320, 1, 7, 1, 2, 6, 3, 1, 6, 4, 1, 4, 2, 1, 69, 1, 4, 3, 3, 1, 14, 3, 1, 3, 1, 10, 2, 694, 2, 4, 21, 1, 1, 1, 3, 3, 10, 2, 1, 2, 2, 1, 3, 5, 1, 3, 9, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`Continued fraction expansion of Sum_{k>=1} (-1)^(k - 1)/k^3 = (3*zeta(3))/4 = 0.901542677369695714...`
	LINKS	`Table of n, a(n) for n=0..99.` `OEIS Wiki, Euler's alternating zeta function` `Eric Weisstein's World of Mathematics, Dirichlet Eta Function` `Wikipedia, Dirichlet Eta Function` `Index entries for continued fractions for constants`
	EXAMPLE	`1/1^3 - 1/2^3 + 1/3^3 - 1/4^3 + 1/5^3 - 1/6^3 +... = 1/(1 + 1/(9 + 1/(6 + 1/(2 + 1/(1 + 1/(1 + 1/...)))))).`
	MATHEMATICA	`ContinuedFraction[(3 Zeta[3])/4, 100]`
	CROSSREFS	`Cf. A013631, A197070.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`Ilya Gutkovskiy, Feb 26 2016`
	STATUS	`approved`

A343244

Position of the first occurrence of an element in the continued fraction of zeta(n) which is larger than the second element.

+10
0

5, 4, 8, 14, 10, 63, 120, 79, 1270, 779, 1749, 3410, 13668, 17704, 20909, 175782, 127426 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`2,1`
	COMMENTS	`a(20) = 111604.` `The corresponding values of the a(n)-th elements are 4, 18, 183, 32, 61, 9283, 462, 1483, 3530, 3484, 10812, 8954, ...`
	LINKS	`Table of n, a(n) for n=2..18.`
	EXAMPLE	`The continued fraction of zeta(3) is [1; 4, 1, 18, 1, 1, ...]. The first element which is larger than 4 is 18 whose position is 4. Therefore, a(3) = 4.`
	MATHEMATICA	`a[n_] := Module[{c = ContinuedFraction[Zeta[n], 10000]}, FirstPosition[c, _?(# > c[[2]] &)][[1]]]; Array[a, 10, 2]`
	CROSSREFS	`Cf. A013697.` `Cf. A013679, A013631, A013680, A013681, A013682, A013683, A013684, A013685.`
	KEYWORD	`nonn,more`
	AUTHOR	`Amiram Eldar, Apr 08 2021`
	STATUS	`approved`

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Last modified September 1 06:23 EDT 2024. Contains 375575 sequences. (Running on oeis4.)