A045708 Primes with first digit 2.

2, 23, 29, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Cf. A000030, subsequence of A208272.

Column k=2 of A262369.

Programs

Haskell

a045708 n = a045708_list !! (n-1)
a045708_list = filter ((== 2) . a000030) a000040_list
-- Reinhard Zumkeller, Mar 16 2012

Magma

[p: p in PrimesUpTo(2300) | Intseq(p)[#Intseq(p)] eq 2]; // Vincenzo Librandi, Aug 08 2014

Mathematica

Select[Table[Prime[n], {n, 3000}], First[IntegerDigits[#]]==2 &] (* Vincenzo Librandi, Aug 08 2014 *)

Python

from sympy import isprime
def agen(limit=float('inf')):
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(1, 10**digits, 2):
      test = adder + i
      if test > limit: return
      if isprime(test): yield test
    digits, adder = digits+1, adder*10
agento = lambda lim: agen(limit=lim)
print(list(agento(2222))) # Michael S. Branicky, Feb 23 2021

Python

from sympy import primepi
def A045708(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(((m:=10**(l:=len(str(x))-1))<<1)-1,x))-primepi(min(3*m-1,x))+sum(primepi(((m:=10**i)<<1)-1)-primepi(3*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

Formula

See A045707 for comments on density of these sequences.

Extensions

More terms from Erich Friedman.

Offset fixed by Reinhard Zumkeller, Mar 15 2012