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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, "-")

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