A045709 Primes with first digit 3.

3, 31, 37, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• easy

Links

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

Crossrefs

Cf. A000040.

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=3 of A262369.

Programs

Magma

[p: p in PrimesUpTo(3300) | Intseq(p)[#Intseq(p)] eq 3]; // Vincenzo Librandi, Aug 08 2014

Mathematica

Select[Table[Prime[n], {n, 4000}], First[IntegerDigits[#]]==3 &] (* Vincenzo Librandi, Aug 08 2014 *)

PARI

isok(n) = isprime(n) && (digits(n, 10)[1] == 3) \\ Michel Marcus, Jun 08 2013

Python

from itertools import chain, count, islice
from sympy import primerange
def A045709_gen(): # generator of terms
    return chain.from_iterable(primerange(3*(m:=10**l),m<<2) for l in count(0))
A045709_list = list(islice(A045709_gen(),40)) # Chai Wah Wu, Dec 07 2024

Python

from sympy import primepi
def A045709(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(3*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min((m<<2)-1,x))+sum(primepi(3*(m:=10**i)-1)-primepi((m<<2)-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

Extensions

More terms from Erich Friedman.