User:Jaume Oliver Lafont/BBP

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(2) = \frac{1}{64}\sum_{k=0}^\infty \left(\frac{64}{6k+1}-\frac{32}{6k+2}-\frac{8}{6k+3}-\frac{8}{6k+4}+\frac{4}{6k+5}+\frac{1}{6k+6}\right)\frac{1}{64^k} = \sum_{n=1}^\infty \frac{1}{cos(\frac{n\pi}{3})n2^n}} (A176900, A002162)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(3) = \frac{1}{32}\sum_{k=0}^\infty \left(\frac{16}{6k+1} + \frac{24}{6k+2} + \frac{16}{6k+3} + \frac{6}{6k+4} + \frac{1}{6k+5}\right)\frac{1}{64^k} = \sum_{n=1}^\infty \frac{1-cos(\frac{n\pi}{3})}{n2^{n-1}}}

Two series related to A014480

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}\log{(1+\sqrt{2})} = \sum_{k=0}^\infty \frac{1}{(2k+1)2^k}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}\arctan{\frac{1}{\sqrt{2}}} = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)2^k} = \frac{1}{2}\sum_{k=0}^\infty \left(\frac{2}{4k+1}-\frac{1}{4k+3}\right)\frac{1}{4^k}}

A ternary zero relation

The sum of zero relations (75) and (76) (103) and (104) in Bailey can be written in six terms.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 = \sum_{k=0}^\infty \frac{1}{(-27)^k}\left({\frac{9}{6k+1}-\frac{9}{6k+2}-\frac{12}{6k+3}-\frac{3}{6k+4}+\frac{1}{6k+5}}\right)} (check)

This formula is similar to equation (17) in Broadhurst, 1998

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_2 = \frac{4}{81}\sum_{k=0}^\infty \left(-\frac{1}{27}\right)^k\left(\frac{9}{(6k+1)^2}-\frac{9}{(6k+2)^2}-\frac{12}{(6k+3)^2}-\frac{3}{(6k+4)^2}+\frac{1}{(6k+5)^2}\right)}

Obtention of a zero relation of the form 0=P(1,2^20,40,A)

From Bellard's formula [1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = \frac{1}{256}\sum_{k=0}^\infty \frac{1}{(-1024)^k}\left( \frac{512}{20k+2} -\frac{160}{20k+5} -\frac{128}{20k+6}-\frac{8}{20k+10} -\frac{8}{20k+14} -\frac{5}{20k+15} +\frac{2}{20k+18}\right)}

In P notation,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = \frac{1}{2^8}P(1,-2^{10},20,(0,512,0,0,-160,-128,0,0,0,-8,0,0,0,-8,-5,0,0,2,0,0))}

Equivalently,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = \frac{1}{2^{18}}P(1,2^{20},40,(0,2^{19},0,0,-5*2^{15},-2^{17},0,0,0,-2^{13},0,0,0,-2^{13},-5*2^{10},0,0,2^{11},0,0,0,-2^{9},0,0,5*2^5,2^7,0,0,0,2^3,0,0,0,2^3,5,0,0,-2,0,0))}

If Ferguson's formula is written with length 40 and both expressions for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4}} subtracted, the following zero relation is obtained.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 = P(1,2^{20},40,(2^{19},-3*2^{19},2^{18},0,2^{19},3*2^{17},-2^{16},0,2^{15},2^{16},2^{14},0,-2^{13},3*2^{13},2^{14},0,2^{11},-3*2^{11},2^{10},0,-2^9,3*2^9,-2^8,0,-2^9,-3*2^7,2^6,0,-2^5,-2^6,-2^4,0,2^3,-3*2^3,-2^4,0,-2,6,-1,0))\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a[4k]=0\,}

This result can be written as a simple linear combination of the three formulas with the same parameters in [2], namely (eq.67)-(eq.66)-(eq.68) -(eq.96)+(eq.97)-(eq.98).

Obtention of a zero relation of the form 0=P(1,2^12,24,A)

Setting a=4 in equation (3) from [3],

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle atan(1/3)=\frac{1}{2^{11}}P(1,2^{12},8,(2^9,2^8,2^6,0,-8,-4,-1,0))}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{2^{11}}P(1,2^{12},24,3*(0,0,2^9,0,0,2^8,0,0,2^6,0,0,0,0,0,-8,0,0,-4,0,0,-1,0,0,0))}

Setting a=2 in equation (4) from [4],

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle atan(1/3)=\frac{1}{2^{4}}P(1,2^{4},8,(2^3,-2^3,2^2,0,-2,2,-1,0))}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{2^{12}}P(1,2^{12},24,(2^{11},-2^{11},2^{10},0,-2^9,2^9,-2^8,0,2^7,-2^7,2^6,0,-2^5,2^5,-2^4,0,2^3,-2^3,2^2,0,-2,2,-1,0))}

Subtracting both results,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0=P(1,2^{12},24,(2^{11},-2^{11},-2^{11},0,-2^9,-2^{10},-2^8,0,-2^8,-2^7,2^6,0,-32,32,32,0,8,16,4,0,4,2,-1,0))\,}

This zero relation can also be written as a BBP-type formula of base -2^6 and is a linear combination of three formulas with the same parameters in [5], namely -(eq.61)-4*(eq.63)-4*(eq.65) -(eq.91)-4*(eq.93)-4*(eq.95)

Binary and ternary formulas

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(2^b-1)=\sum_{k=1}^\infty \frac{1}{k}\left(\frac{b}{2^k} - \frac{1}{2^{bk}}\right)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(3^b-1)=\sum_{k=1}^\infty \frac{1}{k}\left(\frac{3b}{3^k}-\frac{b}{9^k}-\frac{1}{3^{bk}}\right)\,}

			(s,p,q,t,r) in Mathar's table 1.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(2)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{1}{2^k}\right)}	[6]		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{2}{3^k}-\frac{1}{9^k}\right)}	[7]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(3)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{2}{2^k}-\frac{1}{4^k}\right)}	[8]	(1,3,2,2,-1/4)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{3}{3^k}-\frac{1}{9^k}\right)}	[9]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(5)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{2}{2^k}+\frac{1}{4^k}-\frac{1}{16^k}\right)}	[10]	(1,5,2,2,1/4)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{4}{3^k}-\frac{1}{81^k}\right)}	[11]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(7)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{3}{2^k}-\frac{1}{8^k}\right)}	[12]	(1,7,3,2,-1/8)	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{5}{3^k}-\frac{1}{9^k}+\frac{1}{27^k}-\frac{1}{729^k}\right)}	[13]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(11)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{3}{2^k}+\frac{1}{4^k}+\frac{1}{32^k}-\frac{1}{1024^k}\right)}	[14]		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{2}\sum_{k=1}^\infty \frac{1}{k}\left( \frac{13}{3^k}-\frac{4}{9^k}-\frac{1}{243^k}\right)\,}	[15]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(13)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{4}{2^k}-\frac{1}{4^k}+\frac{1}{16^k}+\frac{1}{64^k}-\frac{1}{4096^k}\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{3}{2^k}+\frac{1}{4^k}+\frac{1}{8^k}+\frac{1}{16^k}-\frac{1}{4096^k}\right)}	[16] [17]		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left( \frac{7}{3^k}-\frac{2}{9^k}-\frac{1}{27^k}\right)\,}	[18]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(17)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{4}{2^k}+\frac{1}{16^k}-\frac{1}{256^k}\right)}	[19]	(1,17,4,2,1/16)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(19)\,}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^\infty \frac{1}{k}\left(\frac{3}{2^k}+\frac{3}{4^k}+\frac{1}{512^k}-\frac{1}{262144^k}\right)}	[20]

log(23)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle log(23)=\sum_{k=1}^\infty \left(\frac{4}{2^k}+\frac{1}{3^k}-\frac{1}{24^k}\right)}

23 is the smallest prime whose logarithm is not known to have a binary BBP-type formula.

A016646, A067923.

log(1)

From the identity

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1=\frac{(2^6-1)}{(2^3-1)(2^2-1)^2}}

the following zero relation is obtained:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0=\sum_{k=1}^\infty \frac{1}{k}\left(\frac{1}{2^k}-\frac{2}{2^{2k}}-\frac{1}{2^{3k}}+\frac{1}{2^{6k}}\right)}

This can be shown to be formula (62) in the Compendium by Bailey. See also [21], page 186.

A064078(6)=1.

0=atan(1)-atan(1/2)-atan(1/3)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{16}P(1,16,8,(8,8,4,0,-2,-2,-1,0))} (15 in [22])

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{16}P(1,16,8,(0,16,0,0,0,-4,0,0))\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{16}P(1,16,8,(8,-8,4,0,-2,2,-1,0))\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{16}P(1,16,8,(0,0,0,0,0,0,0,0))\,} (NOT 61 in [23])

Primes as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Pi_{i}(2^{s_{i}}-1)^{q_{i}}} (A144755, A161509)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 3=(2^2-1)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 5=\frac{(2^4-1)}{(2^2-1)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 7=(2^3-1)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 11=\frac{(2^{10}-1)}{(2^5-1)(2^2-1)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 13=\frac{(2^{12}-1)(2^2-1)}{(2^6-1)(2^4-1)}=\frac{(2^{12}-1)}{(2^2-1)(2^3-1)(2^4-1)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 17=\frac{(2^8-1)}{(2^4-1)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 19=\frac{(2^{18}-1)}{(2^9-1)(2^2-1)^3}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 31=(2^5-1)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 41=\frac{(2^{20}-1)(2^2-1)^2}{(2^{10}-1)(2^4-1)^2}\,} (M0.xxx)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 43=\frac{(2^{14}-1)}{(2^7-1)(2^2-1)}\,} (M0.xxx)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 73=\frac{(2^9-1)}{(2^3-1)}\,} (M0.xxx)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 127=(2^7-1)\,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 151=\frac{(2^{15}-1)}{(2^5-1)(2^3-1)}\,} (M0.xxx)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 241=\frac{(2^{24}-1)(2^4-1)}{(2^{12}-1)(2^8-1)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 257=\frac{(2^{16}-1)}{(2^8-1)}} (M0.xxx)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 331=\frac{(2^{30}-1)(2^5-1)}{(2^{15}-1)(2^{10}-1)(2^2-1)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 337=\frac{(2^{21}-1)}{(2^7-1)(2^3-1)^2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 683=\frac{(2^{22}-1)}{(2^{11}-1)(2^2-1)}}

(M0.xxx) first given by Richard J. Mathar in his table of integrals (page 27)

See also

Constants/BBP_Series