A045712 Primes with first digit 6.

61, 67, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Crossrefs

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.

Column k=6 of A262369.

Programs

Magma

[p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 6]; // Bruno Berselli, Jul 19 2014

Mathematica

Flatten[Table[Prime[Range[PrimePi[6 * 10^n] + 1, PrimePi[7 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)
Select[Table[Prime[n],{n, 7000}], First[IntegerDigits[#]]==6 &] (* Vincenzo Librandi, Aug 08 2014 *)

Python

from itertools import chain, count, islice
from sympy import primerange
def A045712_gen(): # generator of terms
    return chain.from_iterable(primerange(6*(m:=10**l),7*m) for l in count(0))
list(islice(A045712_gen(),40)) # Chai Wah Wu, Dec 08 2024

Python

from sympy import primepi
def A045712(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.