A088004 "Truncated Mertens function": values of -1 at primes are left out, that is, summatory Moebius when argument runs through nonprimes.

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 7, 7, 7, 8, 9, 10, 10, 10, 11, 12, 12, 12, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 16, 16, 16, 16, 17, 17, 17, 18, 17, 17, 17, 18, 17, 17, 17, 17, 18, 18, 18, 19, 18, 18, 18, 18

Offset: 1

Author

Labos Elemer, Oct 14 2003

Keywords

• nonn

Comments

Since the principal source of negative excursions of the Mertens function is here eliminated, most probably this sequence increases ad infinitum albeit non-monotonically; it decreases at squarefree numbers with an odd number of prime divisors, e.g., 30 and 42.

Positions of records of a(n) are in A030229. - Michael De Vlieger, May 15 2017

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A002321, A008683, A030229.

Programs

Mathematica

mer[x_] := mer[x-1]+MoebiusMu[x]; mer[0]=0; $RecursionLimit=1000; Table[mer[w]+PrimePi[w], {w, 1, 256}]
(* Second program: *)
Accumulate@ Array[MoebiusMu@ # + Boole[PrimeQ@ #] &, 81] (* Michael De Vlieger, May 15 2017 *)

Formula

a(n) = A002321(n) - (-1)*pi(n) = A002321(n) + A000720(n).