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

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

Offset: 1

Author

Reinhard Zumkeller, Nov 13 2001

Keywords

• base
• nonn

Comments

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.

The "begins with 9" sequence is too dull to include. - N. J. A. Sloane

Note that the primes do not satisfy Benford's law (see A000040). - N. J. A. Sloane, Feb 08 2017

Examples

			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).
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Benford's Law
• Index entries for sequences related to Benford's law

Crossrefs

Cf. A065681, A065682, A065683, A065684, A065685, A065686, A065687, 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.

Programs

Mathematica

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

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