A077677 Squarefree numbers beginning with 1.

1, 10, 11, 13, 14, 15, 17, 19, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 170, 173, 174, 177

Offset: 1

Author

Amarnath Murthy, Nov 16 2002

Keywords

• base
• easy
• nonn

Comments

Intersection of A005117 and A131835. - Michel Marcus, Sep 14 2013

Lower density is 3/(5*Pi^2), upper density is 10/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

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

Crossrefs

Cf. A077678, A077679, A077680, A077681, A077682, A077683, A077684, A077685.

Programs

Mathematica

Select[Range[177], First[IntegerDigits[#]]==1 && SquareFreeQ[#] &] (* Jayanta Basu, May 23 2013 *)

PARI

is(n)=n>0 && digits(n)[1]==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

PARI

list(lim)=my(v=List([1])); for(d=1,#Str(lim\=1)-1, my(D=10^d); forsquarefree(n=D,min(2*D,lim), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 10 2023

Python

from math import isqrt
from sympy import mobius
def A077677(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
    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<1 else h(2*10**((l:=len(s:=str(x)))-2)-1)-g((m:=10**(l-1))-1)+(g(x) if s[0]=='1' else g((m<<1)-1))
    def f(x): return n+x-h(x)
    return bisection(f,n,n) # Chai Wah Wu, May 06 2025

Extensions

Corrected and extended by Sascha Kurz, Jan 28 2003