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A077685 Squarefree numbers beginning with 9.

%I A077685 #28 Jul 27 2025 10:11:56
%S A077685 91,93,94,95,97,901,902,903,905,906,907,910,911,913,914,915,917,919,
%T A077685 921,922,923,926,929,930,933,934,935,937,938,939,941,942,943,946,947,
%U A077685 949,951,953,955,957,958,959,962,965,966,967,969,970,971,973,974,977,978
%N A077685 Squarefree numbers beginning with 9.
%C A077685 Intersection of A005117 and A217402. - _Michel Marcus_, Sep 14 2013
%C A077685 Lower density is 1/(15*Pi^2), upper density is 2/(3*Pi^2). - _Charles R Greathouse IV_, Nov 05 2017
%H A077685 Amiram Eldar, <a href="/A077685/b077685.txt">Table of n, a(n) for n = 1..10000</a>
%t A077685 Select[Range[1000],SquareFreeQ[#]&&First[IntegerDigits[#]]==9&] (* _Harvey P. Dale_, Dec 15 2013 *)
%o A077685 (PARI) isok(n) = issquarefree(n) && (digits(n)[1] == 9); \\ _Michel Marcus_, Sep 14 2013
%o A077685 (Python)
%o A077685 from math import isqrt
%o A077685 from sympy import mobius
%o A077685 def A077685(n):
%o A077685     def bisection(f,kmin=0,kmax=1):
%o A077685         while f(kmax) > kmax: kmax <<= 1
%o A077685         kmin = kmax >> 1
%o A077685         while kmax-kmin > 1:
%o A077685             kmid = kmax+kmin>>1
%o A077685             if f(kmid) <= kmid:
%o A077685                 kmax = kmid
%o A077685             else:
%o A077685                 kmin = kmid
%o A077685         return kmax
%o A077685     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A077685     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)
%o A077685     def f(x): return n+x-h(x)
%o A077685     return bisection(f,n,n) # _Chai Wah Wu_, May 06 2025
%Y A077685 Cf. A077677, A077678, A077679, A077680, A077681, A077682, A077683, A077684.
%K A077685 base,easy,nonn
%O A077685 1,1
%A A077685 _Amarnath Murthy_, Nov 16 2002
%E A077685 More terms from _Sascha Kurz_, Jan 28 2003