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

1,1

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A043803 A043811 A043820 * A129822 A321998 A351624

Adjacent sequences: A077682 A077683 A077684 * A077686 A077687 A077688

Keyword

base,easy,nonn

Author

Amarnath Murthy, Nov 16 2002

Extensions

More terms from Sascha Kurz, Jan 28 2003

Status

approved