A063035 Number of integers m <= 10^n that contain a square factor (i.e., belong to A013929).

3, 39, 392, 3917, 39206, 392074, 3920709, 39207306, 392072876, 3920729058, 39207289720, 392072897726, 3920728981706, 39207289814053, 392072898145897, 3920728981459595

Offset: 1

Author

Robert G. Wilson v, Aug 02 2001

Keywords

• nonn

Comments

Note that "containing a square factor" (A013929) is different from "squareful" (A001694).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..36

Crossrefs

Cf. A001694, A013929, A229099.

For the complementary counts see A053462 and A071172.

Programs

Mathematica

f[n_] := Sum[-MoebiusMu[i]Floor[n/i^2], {i, 2, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}]

PARI

{ default(realprecision, 50); for (n=1, 100, t=10^n - 1; a=10^n - sum(k=1, sqrt(t), moebius(k)*floor(t/k^2)); write("b063035.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 16 2009

Python

from math import isqrt
from sympy import mobius
def A063035(n): return (m:=10**n)-sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # Chai Wah Wu, Jul 20 2024

Formula

Limit_{n->oo} a(n)/10^n = A229099. - Robert G. Wilson v, Aug 12 2014

a(n) = 10^n - A071172(n). - Amiram Eldar, Mar 10 2024

Extensions

More terms from Harry J. Smith, Aug 16 2009

Edited (with a more precise definition and a new value for a(1)) by N. J. A. Sloane, Aug 06 2012. As a result of this change, the programs probably now give the wrong value for a(1). The source of the trouble was the ambiguous meaning of squareful - the official definition of squareful is A001694.