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A261208 Terms of the Leibniz formula (as Euler product) that generate successively better approximations to Pi. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 16 17:38 EDT 2022. Contains 353719 sequences. (Running on oeis4.)