A070237 Numbers k such that the sign of core(k)-phi(k) is not equal to 2*mu(k)^2-1, where core(k) is the squarefree part of k.

1, 420, 660, 780, 840, 1320, 1560, 4620, 5460, 7140, 7980, 8580, 9240, 9660, 10920, 11220, 12012, 12180, 12540, 13020, 13260, 14280, 14820, 15180, 15540, 15708, 15960, 17160, 17220, 17556, 17940, 18060, 18564, 19140, 19320, 19380, 19740

Offset

1,2

Comments

Terms > 1 seem to be multiples of 3. For almost all k, sign(core(k)-phi(k)) = 2*mu(k)^2-1 = 2*A008683(k)^2-1.

From Amiram Eldar, Nov 21 2024: (Start)

1 together with nonsquarefree numbers (A013929) k such that core(k) > phi(k).

If k > 1 is term and m is a squarefree number coprime to k, then k*m is also a term.

The least term above 1 that is not a multiple of 3 is 148728580 = 2^2 * 5 * 7 * 11 * 13 *17 *19 * 23.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 1, 5, 14, 236, 1866, 19480, 196284, 1961242, 19546610, 195387874, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00195..., and the constant C (the reciprocal of the density) in the Formula section is larger than 500 and does not equal 420. (End)

Links

Frank M Jackson, Table of n, a(n) for n = 1..10000

Formula

a(n) = C*n + O(n), with C a constant conjectured to be a(2) = 420.

Mathematica

core[n_] := Module[{m, fac=Select[FactorInteger[n], OddQ[#[[2]]] &]}, If[! SquareFreeQ[n], Times@@Table[fac[[m]][[1]], {m, Length[fac]}], n]]; checkQ[n_] :=   Module[{a=Abs[Sign[core[n]-EulerPhi[n]]-2*MoebiusMu[n]^2+1]}, If[a>0, True, False]]; Select[Range[25000], checkQ] (* Frank M Jackson, Jun 22 2017 *)

Prog

(PARI) for(n=1, 25000, if(abs(sign(core(n)-eulerphi(n))-2*moebius(n)^2+1)>0, print1(n, ", ")))
(PARI) is(k) = {my(f = factor(k)); (core(f) > eulerphi(f)) != issquarefree(f); } \\ Amiram Eldar, Nov 21 2024

Crossrefs

Cf. A000010, A005117, A007913, A008683.

See A013929 for another interpretation.

Sequence in context: A350374 A189982 A376374 * A305416 A156687 A191934

Adjacent sequences: A070234 A070235 A070236 * A070238 A070239 A070240

Keyword

easy,nonn,changed

Author

Benoit Cloitre, May 08 2002

Extensions

Comment and Pari code corrected by Chris Boyd, Mar 08 2014

Status

approved