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

1,2

Links

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

Wikipedia, Euler Product

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).

Example

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).

Prog

(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
(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 $FN(Pi-a, "-")<d d
. . . . s d=$FN(Pi-a, "-")
. . . . w !, n ; Steven Lubars, Aug 14 2015

Crossrefs

Cf. A065091, A076342.

Sequence in context: A049931 A335436 A058983 * A010375 A095028 A011246

Adjacent sequences: A261205 A261206 A261207 * A261209 A261210 A261211

Keyword

nonn

Author

Steven Lubars, Aug 11 2015

Status

approved