A045713 Primes with first digit 7.

7, 71, 73, 79, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base

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=7 of A262369.

Programs

Magma

[p: p in PrimesUpTo(7300) | Intseq(p)[#Intseq(p)] eq 7]; // Vincenzo Librandi, Aug 08 2014

Mathematica

Select[ Table[ Prime[ n ], {n, 1000} ], First[ IntegerDigits[ # ]]==7& ]

Python

from itertools import chain, count, islice
from sympy import primerange
def A045713_gen(): # generator of terms
    return chain.from_iterable(primerange(7*(m:=10**l),m<<3) for l in count(0))
A045713_list = list(islice(A045713_gen(),40)) # Chai Wah Wu, Dec 08 2024

Python

from sympy import primepi
def A045713(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(7*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min((m<<3)-1,x))+sum(primepi(7*(m:=10**i)-1)-primepi((m<<3)-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.

Corrected by Jud McCranie, Jan 03 2001

a(13)=757 added from Vincenzo Librandi, Aug 08 2014