This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A070237 #27 Nov 21 2024 09:05:39
%S A070237 1,420,660,780,840,1320,1560,4620,5460,7140,7980,8580,9240,9660,10920,
%T A070237 11220,12012,12180,12540,13020,13260,14280,14820,15180,15540,15708,
%U A070237 15960,17160,17220,17556,17940,18060,18564,19140,19320,19380,19740
%N 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.
%C A070237 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.
%C A070237 From _Amiram Eldar_, Nov 21 2024: (Start)
%C A070237 1 together with nonsquarefree numbers (A013929) k such that core(k) > phi(k).
%C A070237 If k > 1 is term and m is a squarefree number coprime to k, then k*m is also a term.
%C A070237 The least term above 1 that is not a multiple of 3 is 148728580 = 2^2 * 5 * 7 * 11 * 13 *17 *19 * 23.
%C A070237 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)
%H A070237 Frank M Jackson, <a href="/A070237/b070237.txt">Table of n, a(n) for n = 1..10000</a>
%F A070237 a(n) = C*n + O(n), with C a constant conjectured to be a(2) = 420.
%t A070237 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 *)
%o A070237 (PARI) for(n=1,25000,if(abs(sign(core(n)-eulerphi(n))-2*moebius(n)^2+1)>0,print1(n,",")))
%o A070237 (PARI) is(k) = {my(f = factor(k)); (core(f) > eulerphi(f)) != issquarefree(f);} \\ _Amiram Eldar_, Nov 21 2024
%Y A070237 Cf. A000010, A005117, A007913, A008683.
%Y A070237 See A013929 for another interpretation.
%K A070237 easy,nonn
%O A070237 1,2
%A A070237 _Benoit Cloitre_, May 08 2002
%E A070237 Comment and Pari code corrected by _Chris Boyd_, Mar 08 2014