A274982 - OEIS

%I #67 May 11 2019 18:34:47

%S 1,22,203,1071,29354,245891,14959260,14959260,146023209,1178930480,

%T 20735515065,121559317130,4416249685106,37826529360487,

%U 155364605873808,2291095531474075,27417981382118579,154501831890087986,2116782166626093033,13809261875873749757

%N a(n) is the number of terms required in the Basel Problem, i.e., Sum_{m >= 1} 1/m^2, for the first appearance of n correct digits in the decimal expansion of Pi^2/6 to occur.

%C a(n) = round(1/(floor((1/6)Pi^2 * 10^(n-1))/10^(n-1))) for all n up to at least n=1000 (and it can be shown that this formula almost certainly holds for all n beyond that; see A126809 for a similar problem). - _Jon E. Schoenfield_, Nov 06 2016, Nov 12 2016

%H Jon E. Schoenfield, <a href="/A274982/b274982.txt">Table of n, a(n) for n = 1..1000</a>

%H Ed Sandifer, <a href="http://eulerarchive.maa.org/hedi/HEDI-2003-12.pdf">How Euler Did It: Estimating the Basel Problem</a>, MAA Online (2003).

%e a(2) = 22 because 22 terms (Sum_{m = 1..22} 1/m^2) are required for the first two decimal digits of Pi^2/6 to occur for the first time.

%o (Perl) use ntheory ":all"; use bignum try=>"GMP"; my ($dig,$sum,$exp) = (0, 0, (Pi(40)**2)/6); $exp =~ s/\.//; for my $m (1 .. 1e9) { $sum += 1/($m*$m); (my $str = $sum) =~ s/\.//; print ++$dig, " $m\n" while length($str) > $dig && index($exp, substr($str,0,$dig+1)) == 0; } # _Dana Jacobsen_, Sep 29 2016

%Y Cf. A013661, A126809.

%K nonn,base

%O 1,2

%A _G. L. Honaker, Jr._, Sep 23 2016

%E a(7)-a(11) from _Dana Jacobsen_, Oct 03 2016