A077685 Squarefree numbers beginning with 9.

91, 93, 94, 95, 97, 901, 902, 903, 905, 906, 907, 910, 911, 913, 914, 915, 917, 919, 921, 922, 923, 926, 929, 930, 933, 934, 935, 937, 938, 939, 941, 942, 943, 946, 947, 949, 951, 953, 955, 957, 958, 959, 962, 965, 966, 967, 969, 970, 971, 973, 974, 977, 978

Offset: 1

Author

Amarnath Murthy, Nov 16 2002

Keywords

• base
• easy
• nonn

Comments

Intersection of A005117 and A217402. - Michel Marcus, Sep 14 2013

Lower density is 1/(15*Pi^2), upper density is 2/(3*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Links

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

Crossrefs

Cf. A077677, A077678, A077679, A077680, A077681, A077682, A077683, A077684.

Programs

Mathematica

Select[Range[1000],SquareFreeQ[#]&&First[IntegerDigits[#]]==9&] (* Harvey P. Dale, Dec 15 2013 *)

PARI

isok(n) = issquarefree(n) && (digits(n)[1] == 9); \\ Michel Marcus, Sep 14 2013

Python

from math import isqrt
from sympy import mobius
def A077685(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<9 else h(10**(len(s:=str(x))-1)-1)+(g(x)-g(9*10**(len(s)-1)-1) if s[0]=='9' else 0)
    def f(x): return n+x-h(x)
    return bisection(f,n,n) # Chai Wah Wu, May 06 2025

Extensions

More terms from Sascha Kurz, Jan 28 2003