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

Author

Benoit Cloitre, May 08 2002

Keywords

• easy
• nonn

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

Crossrefs

Cf. A000010, A005117, A007913, A008683.

See A013929 for another interpretation.

Programs

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 *)

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

Formula

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

Extensions

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