A274982 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.

1, 22, 203, 1071, 29354, 245891, 14959260, 14959260, 146023209, 1178930480, 20735515065, 121559317130, 4416249685106, 37826529360487, 155364605873808, 2291095531474075, 27417981382118579, 154501831890087986, 2116782166626093033, 13809261875873749757

Offset: 1

Author

G. L. Honaker, Jr., Sep 23 2016

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
• Ed Sandifer, How Euler Did It: Estimating the Basel Problem, MAA Online (2003).

Crossrefs

Cf. A013661, A126809.

Programs

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

Extensions

a(7)-a(11) from Dana Jacobsen, Oct 03 2016