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A045712 Primes with first digit 6.

%I A045712 #31 Dec 08 2024 17:18:49
%S A045712 61,67,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,
%T A045712 691,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,
%U A045712 6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229
%N A045712 Primes with first digit 6.
%H A045712 Vincenzo Librandi, <a href="/A045712/b045712.txt">Table of n, a(n) for n = 1..5000</a>
%t A045712 Flatten[Table[Prime[Range[PrimePi[6 * 10^n] + 1, PrimePi[7 * 10^n]]], {n, 3}]] (* _Alonso del Arte_, Jul 19 2014 *)
%t A045712 Select[Table[Prime[n],{n, 7000}], First[IntegerDigits[#]]==6 &] (* _Vincenzo Librandi_, Aug 08 2014 *)
%o A045712 (Magma) [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 6]; // _Bruno Berselli_, Jul 19 2014
%o A045712 (Python)
%o A045712 from itertools import chain, count, islice
%o A045712 from sympy import primerange
%o A045712 def A045712_gen(): # generator of terms
%o A045712     return chain.from_iterable(primerange(6*(m:=10**l),7*m) for l in count(0))
%o A045712 list(islice(A045712_gen(),40)) # _Chai Wah Wu_, Dec 08 2024
%o A045712 (Python)
%o A045712 from sympy import primepi
%o A045712 def A045712(n):
%o A045712     def bisection(f,kmin=0,kmax=1):
%o A045712         while f(kmax) > kmax: kmax <<= 1
%o A045712         while kmax-kmin > 1:
%o A045712             kmid = kmax+kmin>>1
%o A045712             if f(kmid) <= kmid:
%o A045712                 kmax = kmid
%o A045712             else:
%o A045712                 kmin = kmid
%o A045712         return kmax
%o A045712     def f(x): return n+x+primepi(min(6*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(7*m-1,x))+sum(primepi(6*(m:=10**i)-1)-primepi(7*m-1) for i in range(l))
%o A045712     return bisection(f,n,n) # _Chai Wah Wu_, Dec 08 2024
%Y A045712 For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.
%Y A045712 Column k=6 of A262369.
%K A045712 nonn,base,easy
%O A045712 1,1
%A A045712 _Felice Russo_
%E A045712 More terms from _Erich Friedman_.