

A186311


Least k such that the interval 100k to 100k+99 has exactly n primes.


21



16718, 1559, 3020, 588, 314, 188, 186, 59, 48, 41, 21, 13, 11, 19, 5, 8, 2, 4, 1228537713709
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

It is known that a(21)=1 and a(25)=0. Terms for n = 19, 20, 22, 23, and 24 are unknown. Glaisher tabulates the number of centuries having 0, 1, 2,... primes for numbers up to 9000000. Glaisher's 1883 book is still in print!
a(24) does not exist because the only century having 24 primes is 0 to 99  the same century having 25 primes. From A020497, we see that a range of 101 numbers is required to find 24 primes. Dickson's conjecture implies that a(n) exists for n=18..23.  Charles R Greathouse IV, Feb 24 2011
a(n) for n = 18..23 is greater than 10^10. Ribenboim discusses Dickson's conjecture in two books.  T. D. Noe, Feb 24 2011
a(19) <= 1108851311300675700427.  Donovan Johnson, Feb 28 2011
a(20) <= 394338677302163715754576644.  Tim Johannes Ohrtmann, Aug 27 2015


REFERENCES

James Glaisher, Factor Table for the Sixth Million, Taylor and Francis, London, 1883.
Paulo Ribenboim, The New Book of Prime Number Records, SpringerVerlag NY, 1995, p. 372.
Paulo Ribenboim, The Little Book of Bigger Primes, SpringerVerlag NY, 2004, p. 250.


LINKS

Table of n, a(n) for n=0..18.
Chris Caldwell, Prime Glossary: Dickson's Conjecture
L. E. Dickson, A new extension of Dirichlet's theorem on prime numbers, Messenger of Math., 33 (1904), 155161.


MATHEMATICA

t = Differences[PrimePi[100*Range[0, 20000]]]; Flatten[Table[Position[t, n, 1, 1], {n, 0, 17}]  1]


PROG

(PARI) a(n)=for(k=0, 9e99, if(sum(i=100*k+1, 100*k+99, ispseudoprime(i))==n, return(k))) \\ Charles R Greathouse IV, Feb 24 2011


CROSSREFS

Cf. A038822 (number of primes between 100n and 100n+99).
Cf. A181098 (centuries without primes).
Cf. A186393A186408 (centuries having 1 to 16 primes).
Cf. A186509 (centuries having 17 primes).
Cf. A261571.
Sequence in context: A232449 A260312 A224572 * A181098 A202896 A183727
Adjacent sequences: A186308 A186309 A186310 * A186312 A186313 A186314


KEYWORD

nonn,fini


AUTHOR

T. D. Noe, Feb 22 2011


EXTENSIONS

a(18) from Donovan Johnson, Feb 28 2011


STATUS

approved



