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

Author

Antti Karttunen, Jun 04 2014

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

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

A120992 gives the lengths of runs.

Programs

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

Python

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))

Formula

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