Steven Lubars - cp's OEIS frontend

A261208 Terms of the Leibniz formula (as Euler product) that generate successively better approximations to Pi.

1, 3, 4, 5, 8, 47, 49, 95, 247, 251, 253, 742, 4268, 4270, 4288, 11445, 30123, 30701, 30703, 62592, 62690, 62992, 3535871, 3535872, 3664203, 3664204, 3664214, 3664220, 3665670, 3665696, 3665842, 3665854, 3665866, 3708907, 3708909, 3708913, 3708929, 3708931, 3708935, 3708957, 3708983, 3708985, 3709017

Offset: 1

Steven Lubars, Aug 11 2015

nonn

			Calculating the first 8 terms: c(1)=3, c(2)=3.75, c(3)=3.28125, c(4)=3.0078125, c(5)=3.2584635416, c(6)=3.462117513020833, c(7)=3.289011637369791, c(8)=3.1519694858127165.
In the above sequence, terms 1, 3, 4, 5, and 8 provide successively closer approximations of Pi (whereas approximations 2, 6, and 7 do not).

Steven Lubars, Table of n, a(n) for n = 1..71
Wikipedia, Euler Product

Cf. A065091, A076342.

s Pi=3.141592653589793238,a=3,n=1,d=Pi-a
w !,1
f i=6:6:1e10 d
  s L=i+1**.5\1
  f j=i-1:2:i+1 d
    f k=3:2:L q:'(j#k)
    i j#k d
      s a=a*j/(j#4+j-2),n=n+1
      i $FN(Pi-a,"-")Steven Lubars, Aug 14 2015

nearmul(p) = if (p % 4 == 1, p-1, p+1);
lista(nn) = {print1(lb = 1, ", "); v = 3; ld = abs(Pi-3); for (n=2, nn, np = prime(n+1); v *= np/nearmul(np); if ((nld=abs(Pi-v)) < ld, print1(n, ", "); ld = nld););} \\ Michel Marcus, Aug 14 2015

Pi = 4*b(1)*b(2)*b(3)*... where b(n) is the n-th odd prime (A065091) divided by its nearest multiple of 4.

Let c(n) be the n-th term of the expansion such that c(n) = 4*b(1)*...*b(n). The sequence consists of the values n such that c(n) provides a closer approximation of Pi than previous approximations c(1)...c(n-1).

A166007 Number of ones in the binary representation of the middle member q of the prime triple (p,q,r) with p

Original entry on oeis.org

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

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 03 2009

			For n = 3, (p, q, r) = (11, 13, 17), q = 13
Decimal 13 = Binary 1101
a(3) = Number of ones in 1101 = 3

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A098414, A014499

DigitCount[#,2,1]&/@Transpose[Select[Partition[Prime[Range[1000]],3,1], Last[#]-First[#]==6&]][[2]] (* Harvey P. Dale, Dec 03 2014 *)

More terms from Harvey P. Dale, Dec 03 2014

A166028 a(n) is the starting position of the second occurrence of a string of the initial n decimal digits of e in the decimal expansion of e.

Original entry on oeis.org

4, 62, 918, 5503, 29169, 252474, 7910575, 7910575

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 04 2009

The digits 2 7 1 8 2 ... are labeled 0, 1, 2, 3, 4, ...

The next term, a(9), is greater than 50 million. - Harvey P. Dale, Apr 01 2017

			a(6) is the starting position of the second occurrence of the initial 6 decimals of e. the string '271828' occurs at position 252,474, thus a(6) = 252,474.

Cf. A001113, A081876

Module[{nn=300000,ed},ed=RealDigits[E,10,nn][[1]];Table[ SequencePosition[ ed, Take[ed,n],2][[2]],{n,6}][[All,1]]-1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 01 2017 *)

Two more terms -- a(7) and a(8) -- from Harvey P. Dale, Apr 01 2017

A166008 Number of ones in the binary representation of the average of twin prime pairs.

Original entry on oeis.org

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

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 03 2009

			Third twin prime pair = (11,13) with average 12 = 1100_2, with 2 ones, so a(3)=2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A014574, A014499.

[&+Intseq(p+1,2):p in PrimesUpTo(3000)|IsPrime(p+2)]; // Marius A. Burtea, Dec 19 2019

seq={1}; Do[If[And @@ PrimeQ[6n + {-1, 1}], AppendTo[seq, DigitCount[6n, 2, 1]]], {n, 1, 600}]; seq (* Amiram Eldar, Dec 19 2019 *)
DigitCount[#,2,1]&/@(Mean/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]- #[[1]] == 2&]) (* Harvey P. Dale, Dec 12 2021 *)

a(n) = A000120(A014574(n)). - Michel Marcus, Dec 19 2019

More terms from Amiram Eldar, Dec 19 2019

A166006 Distance from the origin using the binary expansion of Pi to walk the number line: Start at the origin; subtract one for each '0' digit, and add one for each '1' digit.

Original entry on oeis.org

1, 2, 1, 0, 1, 0, -1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 0, 1, 0, -1, -2, -1, -2, -3, -4, -5, -4, -5, -4, -3, -4, -3, -4, -5, -6, -5, -4, -5, -6, -7, -8, -7, -8, -9, -10, -9, -8, -9, -8, -9, -10, -9, -8, -9, -10, -11, -10, -11, -12, -11, -10, -11

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 03 2009

Of the first 10^10 terms, 5738590822 are positive and 4261262135 are negative. - Hans Havermann, Nov 27 2016

			The first five digits of the expansion are 1, 1, 0, 0, 1.
Starting at 0, we get 0 + 1 + 1 - 1 - 1 + 1 = 1, so a(5) = 1.

Hans Havermann, Table of n, a(n) for n = 1..10000
Hans Havermann, A walk in base-two pi

Cf. A004601, A039624 (indices of zero), A278737 (record maxima), A278738 (record minima), A369900.

a(n) = Sum_{k=1..n} (2*b(k) - 1), where b(n) is the n-th binary digit of Pi.

cp's OEIS Frontend

User: Steven Lubars

A261208 Terms of the Leibniz formula (as Euler product) that generate successively better approximations to Pi.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

MUMPS

PARI

Formula

A166007 Number of ones in the binary representation of the middle member q of the prime triple (p,q,r) with p

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A166028 a(n) is the starting position of the second occurrence of a string of the initial n decimal digits of e in the decimal expansion of e.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A166008 Number of ones in the binary representation of the average of twin prime pairs.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A166006 Distance from the origin using the binary expansion of Pi to walk the number line: Start at the origin; subtract one for each '0' digit, and add one for each '1' digit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula