1,2

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

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).

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.

(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

Cf. A013661, A126809.

Sequence in context: A230896 A031202 A376946 * A144249 A280475 A010828

Adjacent sequences: A274979 A274980 A274981 * A274983 A274984 A274985

nonn,base

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

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

approved