A045712 - OEIS

%I #31 Dec 08 2024 17:18:49

%S 61,67,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,

%T 691,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,

%U 6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229

%N Primes with first digit 6.

%H Vincenzo Librandi, <a href="/A045712/b045712.txt">Table of n, a(n) for n = 1..5000</a>

%t Flatten[Table[Prime[Range[PrimePi[6 * 10^n] + 1, PrimePi[7 * 10^n]]], {n, 3}]] (* _Alonso del Arte_, Jul 19 2014 *)

%t Select[Table[Prime[n],{n, 7000}], First[IntegerDigits[#]]==6 &] (* _Vincenzo Librandi_, Aug 08 2014 *)

%o (Magma) [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 6]; // _Bruno Berselli_, Jul 19 2014

%o (Python)

%o from itertools import chain, count, islice

%o from sympy import primerange

%o def A045712_gen(): # generator of terms

%o     return chain.from_iterable(primerange(6*(m:=10**l),7*m) for l in count(0))

%o list(islice(A045712_gen(),40)) # _Chai Wah Wu_, Dec 08 2024

%o (Python)

%o from sympy import primepi

%o def A045712(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     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     return bisection(f,n,n) # _Chai Wah Wu_, Dec 08 2024

%Y 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 Column k=6 of A262369.

%K nonn,base,easy

%O 1,1

%A _Felice Russo_

%E More terms from _Erich Friedman_.