cp's OEIS Frontend

A172436 Smallest m such that the Moebius function takes successively, from m, n values 1,0,1,0,... ending with 1 or 0.

%I A172436 #10 Aug 18 2025 12:11:54
%S A172436 1,15,55,159,411,411,411,3647,15243,15243,15243,113343,1133759,
%T A172436 1133759,1133759,29149139
%N A172436 Smallest m such that the Moebius function takes successively, from m, n values 1,0,1,0,... ending with 1 or 0.
%C A172436 It's easy to prove that a(n) for n >= 17 does not exist, because in all sequences of 17 consecutive numbers such that the first is squarefree, there are necessarily two numbers r, s where 9 divides r and s, so Moebius(r) = Moebius(s) = 0 with r - s odd.
%D A172436 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
%D A172436 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
%D A172436 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
%D A172436 Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius function. Experiment. Math. 5 (1996), no. 4, 291-295.
%D A172436 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
%H A172436 Ed Pegg Jr., <a href="https://www.mathpuzzle.com/MAA/02-Mobius%20Function/mathgames_11_03_03.html">The Mobius function (and squarefree numbers)</a>
%H A172436 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/TABLES/aaaFArit/MobiusMe.htm">Nombres de Moebius et de Mertens</a>
%H A172436 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>.
%e A172436 a(3) = 55 since Moebius(55) = 1, Moebius(56) = 0, Moebius(57) = 1, and this pattern does not occur for any smaller value of n.
%o A172436 (PARI) a(n)=local(ok,m);m=1;while(1,ok=1;for(k=1,n,if(moebius(m+k-1)!=k%2,ok=0;break));if(ok,return(m));m++)
%Y A172436 Moebius (or Mobius) function mu(n): A008683, A002321, A045882.
%K A172436 nonn,fini,full
%O A172436 1,2
%A A172436 _Michel Lagneau_, Feb 02 2010
%E A172436 Edited by _Franklin T. Adams-Watters_, May 15 2010