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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 (list; graph; refs; listen; history; text; internal format)
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^(d-1) > 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

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Last modified June 17 17:05 EDT 2021. Contains 345085 sequences. (Running on oeis4.)