A073517 Number of primes less than 10^n with initial digit 1.

0, 4, 25, 160, 1193, 9585, 80020, 686048, 6003530, 53378283, 480532488, 4369582734, 40063566855, 369893939287, 3435376839800, 32069022099022, 300694113015105, 2830466318006780, 26735673312004455, 253315661161665338, 2406763761677705769, 22923886160712831134, 218839439542390117580

Offset

1,2

Links

Table of n, a(n) for n=1..23.

Chris K. Caldwell, How Many Primes Are There?

Xavier Gourdon & Pascal Sebah, Counting the number of primes

Henri Lifchitz, Parity of Pi(n)

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Formula

a(n) = Sum_{k=0..n-1} pi(2*10^k-1) - pi(10^k-1). - Andrew Howroyd, Dec 15 2024

Example

a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 1 (11, 13, 17 and 19).

Mathematica

f[n_] := f[n] = PrimePi[2*10^n] - PrimePi[10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 13}]

Prog

(PARI) a(n, d=1)=sum(k=0, n-1, primepi((d+1)*10^k-1) - primepi(d*10^k-1)) \\ Andrew Howroyd, Dec 15 2024

Crossrefs

Cf. A000720 (pi), A073509 to A073517, their sum is A006880.

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: A079750 A195510 A264775 * A208264 A357162 A184755

Adjacent sequences: A073514 A073515 A073516 * A073518 A073519 A073520

Keyword

base,hard,nonn,changed

Author

Shyam Sunder Gupta, Aug 14 2002

Extensions

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(21)-a(22) added by David Baugh, Mar 21 2015

a(23) from Chai Wah Wu, Sep 18 2018

Offset corrected by Andrew Howroyd, Dec 15 2024

Status

approved