A084236 a(n) = M(2^n), where M(n) is Mertens's function, A002321.

1, 0, -1, -2, -1, -4, -1, -2, -1, -4, -4, 7, -19, 22, -32, 26, 14, -20, 24, -125, 257, -362, 228, -10, 211, -1042, 329, 330, -1703, 6222, -10374, 9569, 1814, -10339, -3421, 8435, 38176, -28118, 38729, -135944, 101597, 15295, -169338, 259886, -474483, 1726370, -3554573

Offset: 0

Author

Robert G. Wilson v, May 15 2003

Keywords

• sign

Links

• Chai Wah Wu and Amiram Eldar, Table of n, a(n) for n = 0..75 (terms 0..73 from Hurst's paper added by Chai Wah Wu, terms 74..75 from Helfgott and Thompson's paper added by Amiram Eldar)
• Harald A. Helfgott and Lola Thompson, Summing mu(n): a faster elementary algorithm, arXiv:2101.08773 [math.NT], 2021.
• Greg Hurst, Computations of the Mertens function and improved bounds on the Mertens conjecture, Mathematics of Computation, Vol. 87, No. 310 (2018), pp. 1013-1028; arXiv preprint, arXiv:1610.08551 [math.NT], 2016-2017.

Crossrefs

Cf. A000079, A002321, A008683.

Programs

Mathematica

s = 0; i = 1; Do[ While[i <= 2^n, s = s + MoebiusMu[i]; i++ ]; Print[s], {n, 0, 50}]

PARI

a(n) = sum(k=1, 2^n, moebius(k)) \\ Indranil Ghosh, Mar 15 2017

Formula

a(n) = A002321(2^n).

a(n) = Sum_{k=1..2^n} mu(k), where mu = Moebius function (A008683).

Extensions

a(31)-a(46) from Hurst's paper (copied by Charles R Greathouse IV, Oct 15 2018)