cp's OEIS Frontend

A045707 Primes with first digit 1.

%I A045707 #52 Dec 08 2024 09:56:07
%S A045707 11,13,17,19,101,103,107,109,113,127,131,137,139,149,151,157,163,167,
%T A045707 173,179,181,191,193,197,199,1009,1013,1019,1021,1031,1033,1039,1049,
%U A045707 1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151
%N A045707 Primes with first digit 1.
%C A045707 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
%C A045707 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
%C A045707 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
%H A045707 Vincenzo Librandi, <a href="/A045707/b045707.txt">Table of n, a(n) for n = 1..5000</a>
%H A045707 Daniel I. A. Cohen and Talbot M. Katz, <a href="http://dx.doi.org/10.1016/0022-314X(84)90061-1">Prime numbers and the first digit phenomenon</a>, J. Number Theory 18 (1984), 261-268.
%t A045707 Select[Table[Prime[n], {n, 500}], First[IntegerDigits[#]] == 1 &]
%t A045707 Flatten[Table[Prime[Range[PrimePi[10^n] + 1, PrimePi[2 * 10^n]]], {n, 3}]] (* _Alonso del Arte_, Jul 18 2014 *)
%o A045707 (Magma) [p: p in PrimesUpTo(10^4) | IsOne(Intseq(p)[#Intseq(p)])]; // _Bruno Berselli_, Jul 19 2014
%o A045707 (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
%o A045707 (Python)
%o A045707 from itertools import chain, count, islice
%o A045707 from sympy import primerange
%o A045707 def A045707_gen(): # generator of terms
%o A045707     return chain.from_iterable(primerange(m:=10**l,(m<<1)) for l in count(0))
%o A045707 A045707_list = list(islice(A045707_gen(),40)) # _Chai Wah Wu_, Dec 07 2024
%o A045707 (Python)
%o A045707 from sympy import primepi
%o A045707 def A045707(n):
%o A045707     def bisection(f,kmin=0,kmax=1):
%o A045707         while f(kmax) > kmax: kmax <<= 1
%o A045707         while kmax-kmin > 1:
%o A045707             kmid = kmax+kmin>>1
%o A045707             if f(kmid) <= kmid:
%o A045707                 kmax = kmid
%o A045707             else:
%o A045707                 kmin = kmid
%o A045707         return kmax
%o A045707     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))
%o A045707     return bisection(f,n,n) # _Chai Wah Wu_, Dec 07 2024
%Y A045707 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).
%Y A045707 Cf. A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509
%Y A045707 Column k=1 of A262369.
%K A045707 nonn,base,easy
%O A045707 1,1
%A A045707 _Felice Russo_
%E A045707 More terms from _Erich Friedman_.
%E A045707 Cohen-Katz reference from _Victor S. Miller_, Dec 21 2004