A051402 - OEIS

%I #45 Nov 01 2023 09:57:49

%S 1,5,13,31,110,114,197,199,443,659,661,665,1105,1106,1109,1637,2769,

%T 2770,2778,2791,2794,2795,2797,2802,2803,6986,6987,7013,7021,8503,

%U 8506,8507,8509,8510,8511,9749,9822,9823,9830,9831,9833,9857,9861,19043

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

%C From _Torlach Rush_, Oct 11 2018: (Start)

%C For k <= 10^7:

%C - a(n) is squarefree.

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

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

%H Charles R Greathouse IV, <a href="/A051402/b051402.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

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

%p 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:

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

%o (Haskell)

%o import Data.List (elemIndex)

%o import Data.Maybe (fromJust)

%o a051402 = (+ 1) . fromJust . (`elemIndex` ms) where

%o    ms = map (abs . a002321) [1..]

%o -- _Reinhard Zumkeller_, Dec 26 2012

%o (PARI) M(n)=sum(k=1,n,moebius(k));

%o 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

%o (PARI) M(n) = sum(k=1, n, moebius(k));

%o a(n) = my(k = 1, s = moebius(1)); while (abs(s) != n, k++; s += moebius(k)); k; \\ _Michel Marcus_, Oct 12 2018

%Y Cf. A002321, A008683, A051400, A051401.

%Y Essentially same as A060434 except for initial terms.

%K nonn,nice

%O 1,2

%A _Jud McCranie_