login
Squarefree numbers k for which Q(k) - 6*k/Pi^2 sets a new record minimum, where Q(x) is the number of squarefree numbers up to x.
0

%I #32 Feb 10 2021 08:27:31

%S 1,29,57,173,177,365,370,377,379,381,1090,1865,5578,5590,11326,11333,

%T 11863,11865,11877,11882,12657,13881,32285,32313,32833,32853,32881,

%U 33034,33041,37558,37561,37571,37573,37577,37689,38729,38858,38863,38865,38873,38877

%N Squarefree numbers k for which Q(k) - 6*k/Pi^2 sets a new record minimum, where Q(x) is the number of squarefree numbers up to x.

%C The sequence contains each squarefree integer k where Q(k) - 6*k/Pi^2 is smaller than Q(m) - 6*m/Pi^2 for any 0 < m < k. Where both m and k are squarefree. It is well known that Q(k) is asymptotic to 6*k/Pi^2.

%e Q(29) = 18 and Q(29) - 6*29/Pi^2 = 0.37011... is smaller than Q(1) - 6/Pi^2 = 0.39207...

%t s = Select[Range[50000], SquareFreeQ]; d = 6*s/Pi^2 - Range[Length[s]]; s[[Flatten[Position[d, #][[1]] & /@ Union @ FoldList[Max, d]]]] (* _Amiram Eldar_, Jan 27 2021 *)

%o (PARI) lista(nn) = {my(m=oo, nb=0, x); forsquarefree(n=1, nn, nb++; x = nb - 6*n[1]/Pi^2; if (x < m, m = x; print1(n[1], ", ")););} \\ _Michel Marcus_, Jan 26 2021

%Y Cf. A005117, A275390 (indices of records of |Q(m)-6*m/Pi^2|).

%K nonn

%O 1,2

%A _Adedoyin M. Adegbuyi_, Jan 15 2021

%E More terms from _Jinyuan Wang_, Jan 16 2021