A243349 -id:A243349 - cp's OEIS frontend

A243348 Difference between the n-th squarefree number and n: a(n) = A005117(n) - n.

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, 14, 16, 16, 19, 20, 21, 22, 22, 22, 23, 23, 25, 25, 25, 26, 26, 26, 27, 27, 29, 29, 29, 31, 31, 32, 32, 32, 33, 34, 35, 35, 35, 36, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 42

Offset: 1

Antti Karttunen, Jun 04 2014

nonn

a(n) <= n, as A243351(n) = 2n - A005117(n) goes never negative (please see the plot A005117(n)/n given in the links section).
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.
See also A120992 which gives the run lengths.
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

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Sequence plotted together with A243351 showing how their ratio develops.
Antti Karttunen, Ratio A005117(n)/n plotted in the same way, converging to Pi^2/6.

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

A120992 gives the lengths of runs.

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

from math import isqrt
from sympy import mobius
def A243348(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m-n # Chai Wah Wu, Aug 12 2024

Scheme
```
(define (A243348 n) (- (A005117 n) n))
```

a(n) = A005117(n) - n.
a(n) = A243349(n) - A243289(n).
a(n) = n - A243351(n).
Limit_{n->oo} a(n)/A243351(n) = (Pi^2 - 6)/(12 - Pi^2) = 1.81637833.... - Charles R Greathouse IV, Jun 04 2014
a(n) ~ kn where k = Pi^2/6 - 1 = 0.644934.... - Charles R Greathouse IV, Nov 05 2017

A243289 n minus the index of the greatest prime dividing n-th squarefree number: a(n) = n - A243290(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 4, 3, 3, 6, 8, 5, 5, 10, 10, 7, 11, 8, 16, 9, 16, 15, 19, 12, 17, 20, 14, 24, 15, 21, 16, 25, 17, 29, 27, 26, 20, 20, 28, 34, 36, 23, 34, 40, 25, 25, 35, 43, 43, 28, 38, 29, 46, 40, 45, 32, 51, 47, 44, 52, 36, 36, 56, 37, 61, 50, 39, 39, 64, 58

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

If A005117(n) <= 2n, or equally, if A243351 is always positive, then this sequence is certainly positive as well.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A005117, A006530, A049084, A243290, A243291, A243348, A243349, A243351.

With[{t = Table[PrimePi[FactorInteger[k][[-1, 1]]], {k, Select[Range[120], SquareFreeQ]}]}, Range[Length[t]] - t] (* Amiram Eldar, Mar 04 2024 *)

a(n) = n - A243290(n).

cp's OEIS Frontend

A243348 Difference between the n-th squarefree number and n: a(n) = A005117(n) - n.

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