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Number of primes <= prime(n) which begin with a 1.
8

%I #24 Dec 15 2024 11:01:51

%S 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 13,14,15,16,17,18,19,20,21,22,23,24,25,25,25,25,25,25,25,25,25,25,25,

%U 25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25

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

%C 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 The "begins with 9" sequence is too dull to include. - _N. J. A. Sloane_

%C Note that the primes do not satisfy Benford's law (see A000040). - _N. J. A. Sloane_, Feb 08 2017

%H Harry J. Smith, <a href="/A065680/b065680.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BenfordsLaw.html">Benford's Law</a>

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>

%e 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 Accumulate[If[First[IntegerDigits[#]]==1,1,0]&/@Prime[Range[80]]] (* _Harvey P. Dale_, Jan 22 2013 *)

%o (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 Cf. A065681, A065682, A065683, A065684, A065685, A065686, A065687, A000040.

%Y 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 base,nonn

%O 1,6

%A _Reinhard Zumkeller_, Nov 13 2001