A045707 Primes with first digit 1.

11, 13, 17, 19, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• easy

Comments

Also primes with all divisors starting with digit 1. Complement of A206288 (nonprime numbers with all divisors starting with digit 1) with respect to A206287 (numbers with all divisors starting with digit 1). - Jaroslav Krizek, Mar 04 2012

Cohen and Katz show that the set of primes with first digit 1 has no natural density, but has supernatural/Dirichlet density log_{10} (2) ~= 0.3, the primes with first digit 2 have (supernatural) density log_{10} (3/2) ~= 0.176, ... and the primes with first digit 9 have density log_{10} (10/9) ~= 0.046. This would seem to explain the first digit phenomenon. Note that sum_{k = 1}^9 log_{10} (k+1)/k = 1. - Gary McGuire, Dec 22 2004

Lower density is 1/9, upper density is 5/9. The Dirichlet density, if it exists, is always between the lower and upper density (as it does and is in this case). - Charles R Greathouse IV, Sep 26 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Daniel I. A. Cohen and Talbot M. Katz, Prime numbers and the first digit phenomenon, J. Number Theory 18 (1984), 261-268.

Crossrefs

For primes with initial digit d (1 <= d <= 9) see A045707 (1, this sequence), A045708 (2), A045709 (3), A045710 (4), A045711 (5), A045712 (6), A045713 (7), A045714 (8), A045715 (9).

Cf. A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

Column k=1 of A262369.

Programs

Magma

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

Mathematica

Select[Table[Prime[n], {n, 500}], First[IntegerDigits[#]] == 1 &]
Flatten[Table[Prime[Range[PrimePi[10^n] + 1, PrimePi[2 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 18 2014 *)

PARI

list(lim)=my(v=[]); for(d=1,#digits(lim\=1)-1, v=concat(v,primes([10^d,min(lim,2*10^d-1)]))); v \\ Charles R Greathouse IV, Sep 26 2022

Python

from itertools import chain, count, islice
from sympy import primerange
def A045707_gen(): # generator of terms
    return chain.from_iterable(primerange(m:=10**l,(m<<1)) for l in count(0))
A045707_list = list(islice(A045707_gen(),40)) # Chai Wah Wu, Dec 07 2024

Python

from sympy import primepi
def A045707(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((m:=10**(l:=len(str(x))-1))-1)-primepi(min((m<<1)-1,x))+sum(primepi((m:=10**i)-1)-primepi((m<<1)-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

Extensions

More terms from Erich Friedman.

Cohen-Katz reference from Victor S. Miller, Dec 21 2004