A243348 - OEIS

%I #46 Aug 13 2024 01:55:54

%S 0,0,0,1,1,1,3,3,4,4,4,5,6,7,7,7,9,11,11,11,12,12,12,13,13,13,14,14,

%T 14,16,16,19,20,21,22,22,22,23,23,25,25,25,26,26,26,27,27,29,29,29,31,

%U 31,32,32,32,33,34,35,35,35,36,39,39,39,40,40,40,41,41,41,42,42,42

%N Difference between the n-th squarefree number and n: a(n) = A005117(n) - n.

%C a(n) <= n, as A243351(n) = 2n - A005117(n) goes never negative (please see the plot A005117(n)/n given in the links section).

%C No runs longer than three appear, because there must be at least one gap (cf. A053806) in each range [4k+1 .. 4(k+1)] where no term(s) of A005117 appear.

%C See also A120992 which gives the run lengths.

%C Record values of first differences: a(2) - a(1) = 0, a(4) - a(3) = 1, a(7) - a(6) = 2, a(32) - a(31) = 3, a(151) - a(150) = 4, a(516) - a(515) = 5, a(13392) - a(13391) = 6, a(131965) - a(131964) = 7, a(664314) - a(664313) = 8, a(5392319) - a(5392318) = 9, and a(134453712) - a(134453711) = 11. - _Charles R Greathouse IV_, Nov 05 2017

%H Antti Karttunen, <a href="/A243348/b243348.txt">Table of n, a(n) for n = 1..10001</a>

%H Antti Karttunen, <a href="https://oeis.org/plot2a?name1=A243348&amp;name2=A243351&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Sequence plotted together with A243351 showing how their ratio develops.</a>

%H Antti Karttunen, <a href="https://oeis.org/plot2a?name1=A005117&amp;name2=A000027&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Ratio A005117(n)/n plotted in the same way, converging to Pi^2/6.</a>

%F a(n) = A005117(n) - n.

%F a(n) = A243349(n) - A243289(n).

%F a(n) = n - A243351(n).

%F Limit_{n->oo} a(n)/A243351(n) = (Pi^2 - 6)/(12 - Pi^2) = 1.81637833.... - _Charles R Greathouse IV_, Jun 04 2014

%F a(n) ~ kn where k = Pi^2/6 - 1 = 0.644934.... - _Charles R Greathouse IV_, Nov 05 2017

%o (Scheme) (define (A243348 n) (- (A005117 n) n))

%o (PARI) do(x)=my(v=List([0])); forfactored(n=2,x\1, if(vecmax(n[2][,2])==1, listput(v,n[1]-#v-1))); Vec(v) \\ _Charles R Greathouse IV_, Nov 05 2017

%o (Python)

%o from math import isqrt

%o from sympy import mobius

%o def A243348(n):

%o     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return m-n # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A005117, A053797, A053806, A243351, A243289, A243347.

%Y A120992 gives the lengths of runs.

%K nonn

%O 1,7

%A _Antti Karttunen_, Jun 04 2014