A107079 Minimal number of squared primes in a squarefree gap of length n.

1, 2, 3, 4, 4, 5, 6, 7, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 16, 17, 17, 17, 18, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 30, 31, 32, 32, 32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation

L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - From N. J. A. Sloane, Jan 01

Formula

a(n) = sum{k=0..n-1, moebius_mu(n-k-1) mod 2}.

a(n) = A013928(n+1) + A107078(n).

From Antti Karttunen, Oct 07 2016: (Start)

a(n) = 1 + A013928(n). [Cf. Charles R Greathouse IV's PARI-program.]

For all n >= 1, a(A005117(n)) = n.

(End)

Mathematica

a[n_] := Sum[Boole[SquareFreeQ[k]], {k, 1, n-1}] + 1;
Array[a, 100] (* Jean-François Alcover, Sep 11 2018, from A013928 *)

Prog

(PARI) A107079(n)=1+sum(k=1, n-1, bitand(moebius(k), 1)) \\ Charles R Greathouse IV, Sep 22 2008
(Python)
from math import isqrt
from sympy import mobius
def A107079(n): return 1+sum(mobius(k)*((n-1)//k**2) for k in range(1, isqrt(n-1)+1)) # Chai Wah Wu, Jan 03 2024

Crossrefs

One more than A013928. A left inverse of A005117.

Cf. A045882, A107078.

Sequence in context: A306890 A116549 A268382 * A025528 A255338 A123580

Adjacent sequences: A107076 A107077 A107078 * A107080 A107081 A107082

Keyword

nonn

Author

Paul Barry, May 10 2005

Extensions

New definition from Charles R Greathouse IV, Sep 22 2008

Status

approved