A062982 Numbers n such that Mertens's function of n (A002321) is divisible by phi(n).

1, 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, 537

Offset: 1

Author

Jason Earls, Jul 25 2001

Keywords

• nonn

Comments

Except for the initial term, this sequence is the same as A028442, the n for which Mertens's function M(n) is zero. Because phi(n) >= sqrt(n) and M(n) < sqrt(n) for all known n, phi(n) does not divide M(n), except possibility for some extremely large n. Research project: find the least n > 1 with M(n) not zero and phi(n) divides M(n). - T. D. Noe, Jul 28 2005

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Mertens Conjecture

Crossrefs

Cf. A002321.

Programs

Mathematica

Select[Range[500], Mod[Plus @@ MoebiusMu[Range[#]], EulerPhi[#]] == 0 &] (* Carl Najafi, Aug 17 2011 *)

PARI

M(n)=sum(k=1,n,moebius(k)); j=[]; for(n=1,1500, if(Mod(M(n),eulerphi(n))==0,j=concat(j,n))); j

PARI

{ n=m=0; for (k=1, 10^9, m+=moebius(k); if (m%eulerphi(k)==0, write("b062982.txt", n++, " ", k); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 15 2009