David Brink - cp's OEIS frontend

A260659 Denominators of a BBP-like formula for 4*Pi/sqrt(27).

2, 80, 3584, 1760, 745472, 4456448, 99614720, 265289728, 10905190400, 54492397568, 1065151889408, 1277752770560, 96619584290816, 450799767388160, 8321103999008768, 19017153114013696, 689613692941107200, 3102980143258271744, 55484347409204510720, 30822635849723674624

Offset: 0

David Brink, Nov 13 2015

4*Pi/sqrt(27) = Sum_{n >= 0} (-1/8)^n*(2/(3*n+1)+1/(3*n+2)).

Paolo Xausa, Table of n, a(n) for n = 0..1000
David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants.
David Brink, Nilakantha's accelerated series for pi, Acta Arith. 171 (2015), 293-308.

Cf. A073010, A260658 (numerators).

[Denominator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))): n in [0..60]]; // Vincenzo Librandi, Nov 20 2015

A260659[n_] := Denominator[(-1/8)^n*(2/(3*n + 1) + 1/(3*n + 2))];
Array[A260659, 25, 0] (* Paolo Xausa, Jun 19 2024 *)

a(n) = denominator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))); \\ Michel Marcus, Nov 15 2015

a(n) = denominator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))).

More terms from Michel Marcus, Nov 15 2015

A260658 Numerators of a BBP-like formula for 4*Pi/sqrt(27).

Original entry on oeis.org

5, -7, 23, -1, 41, -25, 59, -17, 77, -43, 95, -13, 113, -61, 131, -35, 149, -79, 167, -11, 185, -97, 203, -53, 221, -115, 239, -31, 257, -133, 275, -71, 293, -151, 311, -5, 329, -169, 347, -89, 365, -187, 383, -49, 401, -205, 419, -107, 437, -223, 455, -29, 473

Offset: 0

David Brink, Nov 13 2015

4*Pi/sqrt(27) = Sum_{n >= 0} (-1/8)^n*(2/(3*n+1)+1/(3*n+2)).

The reduced Collatz function R applied to the numbers 6n+3: a(n) = R(6n+3), where R(k) = (3k+1)/2^r, with r as large as possible, yields an unsigned version of this sequence. - Jonas Kaiser, Jun 17 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants.
David Brink, Nilakantha's accelerated series for pi, Acta Arith. 171 (2015), 293-308.

Cf. A073010, A260659 (denominators).

[Numerator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))): n in [0..60]]; // Vincenzo Librandi, Nov 20 2015

A260658[n_] := Numerator[(-1/8)^n*(2/(3*n + 1) + 1/(3*n + 2))];
Array[A260658, 100, 0] (* Paolo Xausa, Jun 19 2024 *)

a(n) = numerator((-1/8)^n*(2/(3*n+1) + 1/(3*n+2))); \\ Michel Marcus, Nov 15 2015

a(n) = numerator((-1/8)^n*(2/(3*n+1)+1/(3*n+2))).

More terms from Michel Marcus, Nov 15 2015

A234014 Decimal expansion of Sum_{x>=2} 1/((x - 1) sqrt(x)) = Sum_{k>=1} (zeta(k+1/2) - 1).

Original entry on oeis.org

2, 1, 8, 4, 0, 0, 9, 4, 7, 0, 2, 6, 7, 8, 5, 1, 9, 5, 2, 8, 9, 4, 7, 3, 4, 1, 5, 7, 8, 5, 2, 9, 4, 9, 0, 7, 0, 4, 4, 3, 9, 0, 8, 4, 0, 6, 2, 6, 3, 2, 2, 9, 4, 2, 0, 2, 0, 0, 2, 5, 1, 2, 0, 7, 9, 2, 8, 3, 5, 4, 9, 0, 3, 1, 1, 2, 7, 4, 0, 2, 9, 5, 3, 9, 0, 6, 9, 7, 4, 1, 8, 4, 6, 1, 3, 4, 1, 6, 1, 9, 7, 2, 3, 0, 7

Offset: 1

David Brink and Robert G. Wilson v, Dec 18 2013

			2.184009470267851952894734157852949070443908406263229420200251207928354...

David Brink, The Spiral of Theodorus and Sums of Zeta-values at the Half-Integers, The American Mathematical Monthly, Vol 119, No. 9 (November 2012), page 785.

Cf. A105459, A226317, A234015.

RealDigits[ Sum[ Zeta[k + 1/2] - 1, {k, 1, 355}], 10, 111][[1]]

PARI
```
sum(k=1,340,zeta(k+1/2)-1)
```

A234015 K' (K being A105459) = Sum_{k>=0} (Zeta(k+1/2)-1)/(2k+1), negated.

Original entry on oeis.org

1, 8, 2, 6, 5, 0, 7, 8, 1, 0, 8, 5, 8, 4, 7, 8, 5, 5, 8, 8, 1, 5, 7, 6, 5, 4, 0, 6, 1, 3, 0, 3, 2, 1, 9, 7, 3, 0, 9, 9, 4, 9, 1, 4, 8, 4, 9, 4, 3, 4, 9, 0, 6, 6, 8, 3, 2, 2, 9, 0, 1, 3, 6, 3, 7, 7, 6, 4, 9, 9, 2, 7, 1, 8, 3, 8, 7, 3, 5, 8, 4, 6, 4, 7, 9, 7, 3, 1, 3, 6, 2, 1, 5, 8, 3, 2, 8, 9, 9, 4, 2, 0, 4, 7, 1

Offset: 1

David Brink and Robert G. Wilson v, Dec 18 2013

			1.8265078108584785588157654061303219730994914849434906683229013637764992...

David Brink, The Spiral of Theodorus and Sums of Zeta-values at the Half-Integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), page 785.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 663 (constant K').

Cf. A105459, A226317, A234014.

RealDigits[ Sum[ (Zeta[k + 1/2] - 1)/(2 k + 1), {k, 0, 370}], 10,  111][[1]]

PARI
```
sum(k=0,340,(zeta(k+1/2)-1)/(2*k+1))
```

A182009 a(n) = ceiling(sqrt(2nlog(2))+(3-2log(2))/6).

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

David Brink, Apr 06 2012

nonn

This sequence approximates the sequence of solutions to the Birthday Problem, A033810. The two sequences agree for almost all n, i.e., on a set of integers n with density 1.

Gheorghe Coserea, Table of n, a(n) for n = 1..10000
D. Brink, A (probably) exact solution to the Birthday Problem, Ramanujan Journal, 2012, pp 223-238.

Approximates A033810.

seq(ceil((2*n*log(2))^(1/2) + (3-2*log(2))/6), n=1..1000); # Robert Israel, Aug 23 2015

Table[Ceiling[Sqrt[2 n Log[2] + (3 - 2 Log[2])/6]], {n, 82}] (* Michael De Vlieger, Aug 24 2015 *)

a(n) = { ceil((2*n*log(2))^(1/2) + (3-2*log(2))/6) };
apply(n->a(n), vector(84, i, i))  \\ Gheorghe Coserea, Aug 23 2015

A182008 a(n) = ceiling(sqrt(2nlog(2))).

Original entry on oeis.org

2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

David Brink, Apr 06 2012

nonn

This sequence approximates the sequence of solutions to the Birthday Problem, A033810. The two sequences agree on a set of integers n with density (3+2 log 2)/6 = 0.731...

Gheorghe Coserea, Table of n, a(n) for n = 1..10000
D. Brink, A (probably) exact solution to the Birthday Problem, Ramanujan Journal, 2012, pp 223-238.

Approximates A033810.

[Ceiling(Sqrt(2*n*Log(2))): n in [1..100]]; // Vincenzo Librandi, Aug 23 2015

Table[Ceiling[Sqrt[2 n Log[2]]], {n, 100}] (* Vincenzo Librandi, Aug 23 2015 *)

a(n) = { ceil(sqrt(2*n*log(2))) };
apply(n->a(n), vector(88, i, i))  \\ Gheorghe Coserea, Aug 23 2015

A185051 Continued fraction expansion of Hlawka's Schneckenkonstante K = -2.157782...

Original entry on oeis.org

-3, 1, 5, 2, 1, 24, 9, 2, 2, 1, 1, 6, 8, 11, 2, 44, 1, 5, 3, 424, 1, 5, 39, 2, 1, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 4, 2, 1, 1, 15, 2, 3, 3, 3, 2, 1, 45, 15, 10, 16, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 1, 2, 3, 2, 14, 3, 5, 1, 2, 1, 19, 1, 4, 16, 5, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 16, 65, 2, 3, 3, 1, 5, 3, 1, 11, 2, 1, 3, 1, 1, 2, 5, 2, 11, 1, 2, 2, 1, 10, 1, 1, 2

Offset: 0

David Brink, Jan 22 2012

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]

Cf. A105459 for decimal expansion.

A105459 Decimal expansion of Hlawka's Schneckenkonstante K = -2.157782... (negated).

Original entry on oeis.org

2, 1, 5, 7, 7, 8, 2, 9, 9, 6, 6, 5, 9, 4, 4, 6, 2, 2, 0, 9, 2, 9, 1, 4, 2, 7, 8, 6, 8, 2, 9, 5, 7, 7, 7, 2, 3, 5, 0, 4, 1, 3, 9, 5, 9, 8, 6, 0, 7, 5, 6, 2, 4, 5, 5, 1, 5, 4, 8, 9, 5, 5, 5, 0, 8, 5, 8, 8, 6, 9, 6, 4, 6, 7, 9, 6, 6, 0, 6, 4, 8, 1, 4, 9, 6, 6, 9, 4, 2, 9, 8, 9, 4, 6, 3, 9, 6, 0, 8, 9, 8

Offset: 1

David Brink, Jun 13 2011

			-2.157782996659446220929142786829577723504139598607562455...

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

Robert G. Wilson v, Table of n, a(n) for n = 1..16384
David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]
Herbert Kociemba, The Spiral of Theodorus.

Cf. A185051 for continued fraction expansion.

Cf. A072895, A137515, A352741.

evalf(Sum((-1)^k*Zeta(k + 1/2)/(2*k+1), k=0..infinity), 120); # Vaclav Kotesovec, Mar 01 2016

RealDigits[ NSum[(-1)^k*Zeta[k + 1/2]/(2 k + 1), {k, 0, Infinity}, Method -> "AlternatingSigns", AccuracyGoal -> 2^6, PrecisionGoal -> 2^6, WorkingPrecision -> 2^7], 10, 2^7][[1]] (* Robert G. Wilson v, Jul 11 2013 *)

sumalt(k=0,(-1)^k*zeta(k+1/2)/(2*k+1)) \\ M. F. Hasler, Mar 31 2022

Sum_{x=1..n-1} arctan(1/sqrt(x)) = 2*sqrt(n) + K + o(1). [Corrected by M. F. Hasler, Mar 31 2022]

Equals Sum_{k>=0} (-1)^k*zeta(k+1/2)/(2*k+1). - Robert B Fowler, Oct 23 2022

A159456 Discriminant of the 47 imaginary, bicyclic, biquadratic fields with class number 1.

Original entry on oeis.org

144, 225, 256, 400, 441, 576, 576, 784, 1089, 1225, 1600, 1936, 2601, 2704, 3136, 3249, 5776, 5929, 7744, 7744, 8281, 15129, 16641, 17689, 21904, 23104, 29584, 34969, 40401, 43681, 53824, 71289, 71824, 90601, 118336, 182329, 239121, 287296

Offset: 1

David Brink, Apr 12 2009

E. Brown and C. J. Parry, The imaginary bicyclic biquadratic fields with class number 1, J. Reine Angew. Math. 266 (1974), 118-120.

A161891 Primes p with the property that every non-solvable transitive permutation group of degree p is alternating or symmetric.

Original entry on oeis.org

5, 19, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97

Offset: 1

David Brink, Jun 21 2009

nonn

Wielandt mentions only p=5, 19, 47, 59. The rest follows from the classification of finite simple groups.

Wielandt, Finite Permutation Groups, p. 29.
Dixon, Mortimer, Permutation Groups, Springer, 1996, p. 99.

cp's OEIS Frontend

User: David Brink

A260659 Denominators of a BBP-like formula for 4*Pi/sqrt(27).

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A260658 Numerators of a BBP-like formula for 4*Pi/sqrt(27).

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A234014 Decimal expansion of Sum_{x>=2} 1/((x - 1) sqrt(x)) = Sum_{k>=1} (zeta(k+1/2) - 1).

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A234015 K' (K being A105459) = Sum_{k>=0} (Zeta(k+1/2)-1)/(2k+1), negated.

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A182009 a(n) = ceiling(sqrt(2n*log(2))+(3-2*log(2))/6).

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Maple

Mathematica

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A182008 a(n) = ceiling(sqrt(2*n*log(2))).

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Magma

Mathematica

PARI

A185051 Continued fraction expansion of Hlawka's Schneckenkonstante K = -2.157782...

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A105459 Decimal expansion of Hlawka's Schneckenkonstante K = -2.157782... (negated).

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A182009 a(n) = ceiling(sqrt(2nlog(2))+(3-2log(2))/6).

A182008 a(n) = ceiling(sqrt(2nlog(2))).