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A065680 Number of primes <= prime(n) which begin with a 1.

%I A065680 #25 Feb 16 2025 08:32:45
%S A065680 0,0,0,0,1,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,7,8,9,10,11,12,
%T A065680 13,14,15,16,17,18,19,20,21,22,23,24,25,25,25,25,25,25,25,25,25,25,25,
%U A065680 25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25
%N A065680 Number of primes <= prime(n) which begin with a 1.
%C A065680 Considering the frequency of all decimal digits in leading position of prime numbers (A065681 - A065687), we cannot apply Benford's Law. But we observe at 10^e - levels that the frequency for 0 to 9 decreases monotonically, at least in the small range until 10^7.
%C A065680 The "begins with 9" sequence is too dull to include. - _N. J. A. Sloane_
%C A065680 Note that the primes do not satisfy Benford's law (see A000040). - _N. J. A. Sloane_, Feb 08 2017
%H A065680 Harry J. Smith, <a href="/A065680/b065680.txt">Table of n, a(n) for n = 1..1000</a>
%H A065680 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BenfordsLaw.html">Benford's Law</a>
%H A065680 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%e A065680 13 is the second prime beginning with 1: A000040(6) = 13, therefore a(6) = 2. a(664579) = 80020 (A000040(664579) = 9999991 is the largest prime < 10^7).
%t A065680 Accumulate[If[First[IntegerDigits[#]]==1,1,0]&/@Prime[Range[80]]] (* _Harvey P. Dale_, Jan 22 2013 *)
%o A065680 (PARI) lista(n) = { my(a=[p\10^logint(p,10)==1 | p<-primes(n)]); for(i=2, #a, a[i]+=a[i-1]); a} \\ _Harry J. Smith_, Oct 26 2009
%Y A065680 Cf. A065681, A065682, A065683, A065684, A065685, A065686, A065687, A000040.
%Y A065680 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.
%K A065680 base,nonn
%O A065680 1,6
%A A065680 _Reinhard Zumkeller_, Nov 13 2001