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

Author

Steven Lubars, Aug 11 2015

Keywords

• nonn

Examples

			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).
		

Links

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

Crossrefs

Cf. A065091, A076342.

Programs

MUMPS

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

PARI

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

Formula

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).