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

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

• base
• hard
• nonn

Examples

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

Links

• 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)

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.

Programs

Mathematica

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

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

Formula

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

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