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

1,1

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

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 *)

Prog

(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

Crossrefs

Intersection of A005117 and A217397.

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

Sequence in context: A191754 A165862 A339568 * A283598 A041839 A098064

Adjacent sequences: A077677 A077678 A077679 * A077681 A077682 A077683

Keyword

base,easy,nonn

Author

Amarnath Murthy, Nov 16 2002

Extensions

More terms from Sascha Kurz, Jan 28 2003

Status

approved