A051402 Inverse Mertens function: smallest k such that |M(k)| = n, where M(x) is Mertens's function A002321.

1, 5, 13, 31, 110, 114, 197, 199, 443, 659, 661, 665, 1105, 1106, 1109, 1637, 2769, 2770, 2778, 2791, 2794, 2795, 2797, 2802, 2803, 6986, 6987, 7013, 7021, 8503, 8506, 8507, 8509, 8510, 8511, 9749, 9822, 9823, 9830, 9831, 9833, 9857, 9861, 19043

Offset: 1

Author

Jud McCranie

Keywords

• nonn
• nice

Comments

From Torlach Rush, Oct 11 2018: (Start)

For k <= 10^7:

- a(n) is squarefree.

- if a(n) > M(k) then A008683(a(n)) is negative.

- if a(n) = M(k) then A008683(a(n)) is positive. (End)

Examples

			M(31) = -4, smallest one equal to +-4.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Crossrefs

Cf. A002321, A008683, A051400, A051401.

Essentially same as A060434 except for initial terms.

Programs

Haskell

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a051402 = (+ 1) . fromJust . (`elemIndex` ms) where
   ms = map (abs . a002321) [1..]
-- Reinhard Zumkeller, Dec 26 2012

Maple

with(numtheory): k := 0: s := 0: for n from 1 to 20000 do s := s+mobius(n): if abs(s) > k then k := abs(s): print(n); fi; od:

Mathematica

a = s = 0; Do[s = s + MoebiusMu[n]; If[ Abs[s] > a, a = Abs[s]; Print[n]], {n, 1, 20000}]

PARI

M(n)=sum(k=1,n,moebius(k));
print1(1,", "); c=M(1); n=2; while(n<10^3,if(abs(M(n))>c,print1(n,", "); c=abs(M(n))); n++) \\ Derek Orr, Jun 14 2016

PARI

M(n) = sum(k=1, n, moebius(k));
a(n) = my(k = 1, s = moebius(1)); while (abs(s) != n, k++; s += moebius(k)); k; \\ Michel Marcus, Oct 12 2018