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

1,1

Links

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

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 *)

Prog

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

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.

Sequence in context: A141167 A062673 A284291 * A090154 A039390 A043213

Adjacent sequences: A045709 A045710 A045711 * A045713 A045714 A045715

Keyword

nonn,base,easy

Author

Felice Russo

Extensions

More terms from Erich Friedman.

Status

approved