A077680 Squarefree numbers beginning with 4.

41, 42, 43, 46, 47, 401, 402, 403, 406, 407, 409, 410, 411, 413, 415, 417, 418, 419, 421, 422, 426, 427, 429, 430, 431, 433, 434, 435, 437, 438, 439, 442, 443, 445, 446, 447, 449, 451, 453, 454, 455, 457, 458, 461, 462, 463, 465, 466, 467, 469, 470, 471, 473

Offset: 1

Author

Amarnath Murthy, Nov 16 2002

Keywords

• base
• easy
• nonn

Comments

Lower density is 3/(20*Pi^2), upper density is 4/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005117 and A217397.

Cf. A077677, A077678, A077679, A077681, A077682, A077683, A077684, A077685.

Programs

Maple

select(numtheory:-issqrfree, [seq(seq(i,i=4*10^d+1 .. 5*10^d-1),d=1..3)]); # Robert Israel, May 07 2025

Mathematica

Select[Range[499],First[IntegerDigits[#]]==4&&SquareFreeQ[#]&] (* Harvey P. Dale, Apr 24 2018 *)

PARI

is(n)=n>40 && digits(n)[1]==4 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

Python

from functools import lru_cache
from math import isqrt
from sympy import mobius
def A077680(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    @lru_cache(maxsize=None)
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def h(x): return 0 if x<4 else h(5*10**((l:=len(s:=str(x)))-2)-1)-g(4*10**(l-1)-1)+(g(x) if s[0]=='4' else g(5*10**(l-1)-1) if s[0]>'4' else 0)
    def f(x): return n+x-h(x)
    return bisection(f,n,n) # Chai Wah Wu, May 07 2025

Extensions

More terms from Sascha Kurz, Jan 28 2003