A084237 - OEIS

%I #40 Feb 16 2025 08:32:49

%S 1,-1,1,2,-23,-48,212,1037,1928,-222,-33722,-87856,62366,599582,

%T -875575,-3216373,-3195437,-21830254,-46758740,899990187,461113106,

%U -3395895277,-2061910120,62467771689

%N a(n) = M(10^n), where M(n) is Mertens's function.

%H Bernardo Boncompagni, <a href="https://web.archive.org/web/20111009173908/http://mertens.redgolpe.com:80/">Selected values of the Mertens function</a>.

%H Eugene Kuznetsov, <a href="http://arxiv.org/abs/1108.0135">Computing the Mertens function on a GPU</a>, arXiv:1108.0135 [math.NT], 2011.

%H Harald A. Helfgott and Lola Thompson, <a href="https://arxiv.org/abs/2101.08773">Summing mu(n): a faster elementary algorithm</a>, arXiv:2101.08773 [math.NT], 2021.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MertensFunction.html">Mertens Function</a>.

%F Mertens's function: Sum_{k=1..n} mu(k), where mu = Möbius function (A008683).

%F a(n) = A002321(10^n).

%t s = 0; i = 1; Do[ While[i <= 10^n, s = s + MoebiusMu[i]; i++ ]; Print[s], {n, 0, 50}]

%o (Perl) use ntheory ":all"; say mertens(10**$_) for 0..15; # _Dana Jacobsen_, May 22 2015

%Y Cf. A002321, A008683.

%K sign,more

%O 0,4

%A _Robert G. Wilson v_, May 15 2003

%E More terms from _Eric W. Weisstein_, Jun 27 2003

%E a(17) from _Bernardo Boncompagni_, Jul 06 2011

%E Corrected a(17) and added a(18)-a(22) from Eugene Kuznetsov, a(17)-a(19) independently confirmed by _Richard Sladkey_, Aug 28 2012

%E a(21)'s sign correction and a(23) from Helfgott and Thompson (2021) added by _Amiram Eldar_, May 21 2021