A045715 Primes with first digit 9.

97, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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=9 of A262369.

Programs

Magma

[p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 9]; // Bruno Berselli, Jul 19 2014

Magma

[p: p in PrimesInInterval(9*10^n,10^(n+1)), n in [0..3]]; // Bruno Berselli, Aug 08 2014

Mathematica

Flatten[Table[Prime[Range[PrimePi[9 * 10^n] + 1, PrimePi[10^(n + 1)]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)

Python

from itertools import chain, count, islice
def A045715_gen(): # generator of terms
    return chain.from_iterable(primerange(9*(m:=10**l),10*m) for l in count(0))
A045715_list = list(islice(A045715_gen(),40)) # Chai Wah Wu, Dec 08 2024

Python

from sympy import primepi
def A045715(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(9*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(10*m-1,x))+sum(primepi(9*(m:=10**i)-1)-primepi(10*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.