A045711 Primes with first digit 5.

5, 53, 59, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• easy

Comments

Subsequence of A000040.

Links

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

Crossrefs

Column k=5 of A262369.

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.

Programs

Magma

[p: p in PrimesUpTo(5300) | Intseq(p)[#Intseq(p)] eq 5]; // Vincenzo Librandi, Aug 08 2014

Mathematica

Select[Table[Prime[n], {n, 5300}], First[IntegerDigits[#]]==5 &] (* Vincenzo Librandi, Aug 08 2014 *)

Python

from itertools import chain, count, islice
from sympy import primerange
def A045711_gen(): # generator of terms
    return chain.from_iterable(primerange(5*(m:=10**l),6*m) for l in count(0))
A045711_list = list(islice(A045711_gen(),40)) # Chai Wah Wu, Dec 08 2024

Python

from sympy import primepi
def A045711(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(5*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(6*m-1,x))+sum(primepi(5*(m:=10**i)-1)-primepi(6*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.

Leading 5 added by Jaroslav Krizek, Apr 29 2010