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

1,1

Links

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

Mathematica

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

Prog

(PARI) isok(n) = isprime(n) && (digits(n, 10)[1] == 3) \\ Michel Marcus, Jun 08 2013
(Magma) [p: p in PrimesUpTo(3300) | Intseq(p)[#Intseq(p)] eq 3]; // Vincenzo Librandi, Aug 08 2014
(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

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.

Sequence in context: A141177 A154502 A046282 * A090151 A198187 A198019

Adjacent sequences: A045706 A045707 A045708 * A045710 A045711 A045712

Keyword

nonn,base,easy

Author

Felice Russo

Extensions

More terms from Erich Friedman.

Status

approved