cp's OEIS Frontend

A077677 Squarefree numbers beginning with 1.

%I A077677 #28 Jul 27 2025 08:56:45
%S A077677 1,10,11,13,14,15,17,19,101,102,103,105,106,107,109,110,111,113,114,
%T A077677 115,118,119,122,123,127,129,130,131,133,134,137,138,139,141,142,143,
%U A077677 145,146,149,151,154,155,157,158,159,161,163,165,166,167,170,173,174,177
%N A077677 Squarefree numbers beginning with 1.
%C A077677 Intersection of A005117 and A131835. - _Michel Marcus_, Sep 14 2013
%C A077677 Lower density is 3/(5*Pi^2), upper density is 10/(3*Pi^2). - _Charles R Greathouse IV_, Nov 05 2017
%H A077677 Amiram Eldar, <a href="/A077677/b077677.txt">Table of n, a(n) for n = 1..10000</a>
%t A077677 Select[Range[177], First[IntegerDigits[#]]==1 && SquareFreeQ[#] &] (* _Jayanta Basu_, May 23 2013 *)
%o A077677 (PARI) is(n)=n>0 && digits(n)[1]==1 && issquarefree(n) \\ _Charles R Greathouse IV_, Nov 05 2017
%o A077677 (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
%o A077677 (Python)
%o A077677 from math import isqrt
%o A077677 from sympy import mobius
%o A077677 def A077677(n):
%o A077677     def bisection(f,kmin=0,kmax=1):
%o A077677         while f(kmax) > kmax: kmax <<= 1
%o A077677         kmin = kmax >> 1
%o A077677         while kmax-kmin > 1:
%o A077677             kmid = kmax+kmin>>1
%o A077677             if f(kmid) <= kmid:
%o A077677                 kmax = kmid
%o A077677             else:
%o A077677                 kmin = kmid
%o A077677         return kmax
%o A077677     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A077677     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))
%o A077677     def f(x): return n+x-h(x)
%o A077677     return bisection(f,n,n) # _Chai Wah Wu_, May 06 2025
%Y A077677 Cf. A077678, A077679, A077680, A077681, A077682, A077683, A077684, A077685.
%K A077677 base,easy,nonn
%O A077677 1,2
%A A077677 _Amarnath Murthy_, Nov 16 2002
%E A077677 Corrected and extended by _Sascha Kurz_, Jan 28 2003