

A065680


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


8



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
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OFFSET

1,6


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


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


EXAMPLE

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).


MATHEMATICA

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


PROG

(PARI) digitsIn(x)= { local(d); if (x==0, return(1)); d=1 + log(x)\log(10); if (10^d == x, d++, if (10^(d1) > x, d)); return(d) } MSD(x)= { return(x\10^(digitsIn(x)1)) } { a=0; p=2; for (n=1, 1000, q=prime(n); while (p <= q, if(MSD(p) == 1, a++); p=nextprime(p+1)); write("b065680.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 26 2009


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.
Sequence in context: A193669 A065686 A158411 * A093391 A210964 A029135
Adjacent sequences: A065677 A065678 A065679 * A065681 A065682 A065683


KEYWORD

base,nonn


AUTHOR

Reinhard Zumkeller, Nov 13 2001


STATUS

approved



