A186311 -id:A186311 - cp's OEIS frontend

A186405 Numbers k such that there are 13 primes between 100k and 100k + 99.

19, 36, 54, 55, 62, 69, 86, 88, 96, 119, 124, 156, 166, 174, 201, 211, 215, 220, 238, 240, 308, 320, 323, 329, 355, 408, 412, 416, 427, 442, 544, 569, 606, 616, 633, 636, 674, 713, 775, 798, 806, 832, 875, 888, 900, 923, 1098

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 25913910 possible prime patterns for centuries having 13 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			19 is in this sequence because there are 13 primes between 1900 and 1999 (1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997 and 1999).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A038822 (number of primes between 100n and 100n+99), A186311 (first occurrences).

Cf. A181098 (no primes), A186393-A186408 (1 to 16 primes), A186509 (17 primes), A361723 (18 primes).

for(n=1, 1e6, if(sum(k=100*n, 100*(n+1), ispseudoprime(k))==13, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 4e9, if(p>N, if(s==13, print1((N\100)-1, ", ")); s=1; N=100*(p\100+1), s++)) \\ Charles R Greathouse IV, Feb 21 2011

A186406 Numbers k such that there are 14 primes between 100k and 100k + 99.

Original entry on oeis.org

5, 7, 9, 20, 27, 35, 82, 147, 179, 277, 286, 514, 556, 694, 709, 796, 810, 1158, 1363, 1416, 2033, 2173, 2232, 2297, 2660, 3054, 3274, 4508, 4996, 6635, 8194, 8237, 11047, 11467, 12303, 16166, 19543, 19882, 19936

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 20053913 possible prime patterns for centuries having 14 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			5 is in this sequence because there are 14 primes between 500 and 599 (503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593 and 599).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A038822 (number of primes between 100n and 100n+99), A186311 (first occurrences).

Cf. A181098 (no primes), A186393-A186408 (1 to 16 primes), A186509 (17 primes), A361723 (18 primes).

Select[Range[20000],PrimePi[100#+99]-PrimePi[100#]==14&] (* Harvey P. Dale, Jun 20 2021 *)

for(n=1, 1e6, if(sum(k=100*n,100*(n+1), ispseudoprime(k))==14, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 4e9, if(p>N, if(s==14, print1((N\100)-1,", ")); s=1; N=100*(p\100+1),s++)) \\ Charles R Greathouse IV, Feb 21 2011

a(30)-a(39) from Charles R Greathouse IV, Feb 21 2011

A216289 Smallest k in which there are exactly n primes between 10k and 10k+9.

Original entry on oeis.org

20, 9, 2, 4, 0

Offset: 0

V. Raman, Sep 03 2012

Cf. A186311, A032352, A007811, A078494.

mx = 5; t = Table[-1, {mx}]; n = 0; found = 0; While[found < mx, ps = Select[Range[10*n, 10*n + 9], PrimeQ]; len = Length[ps]; If[t[[len + 1]] == -1, t[[len + 1]] = n; found++]; n++]; t (* T. D. Noe, Sep 03 2012 *)

A279862 a(n) = the smallest integer k where there are exactly n primes between 10k and 10k+100.

Original entry on oeis.org

37027, 15590, 3562, 3561, 1881, 1856, 735, 588, 132, 131, 188, 111, 89, 47, 44, 32, 20, 11, 9, 8, 5, 3, 2

Offset: 0

Wolfram Hüttermann, Dec 21 2016

a(25) = 0, and a(24) cannot exist. The same is true with a(k) and k>25. From A020497, we see that a range of 101 numbers is required to find 24 primes. It is an open question if a(23) exists.
From Robert Israel, Jan 18 2017: (Start)
Dickson's conjecture implies that a(23) does exist.
Let Q = 27926129625869590, and R = 614889782588491410 the product of all primes < 50.
Then for any k, the 23 numbers Q+i+k*R for i = 1, 3, 7, 9, 13, 19, 21, 27, 31, 33, 37, 43, 49, 51, 57, 63, 69, 73, 79, 87, 91, 97, 99 have no prime divisors < 50.
Dickson's conjecture would indicate that there are infinitely many k for which these numbers are all prime, and thus there are 23 primes between Q+k*R and Q+k*R+100. (End)
Heuristics suggest a(23) exists (see above) and has between 20 and 30 digits. There are 192 residue classes mod 23# = 223092870 in which a(23) might fall, all of which are 11 mod 30 and either 3 or 4 mod 7. - Charles R Greathouse IV, Jul 12 2017

			For n = 1 there is only one prime between 155900 and 156000: 155921.

Cf. A186311.

For the Dickson conjecture, see A020497.

for n from 1 to 10^5 do
  T[n]:= nops(select(isprime, [$10*n+1 ..10*n+9]))
od:
for k from 1 to 10^5-10 do
   v:= add(T[k+j],j=0..9):
   if not assigned(A[v]) then A[v]:= k fi
od:
seq(A[n],n=0..22); # Robert Israel, Jul 12 2017

Function[s, -1 + Flatten@ Table[FirstPosition[s, n] /. k_ /; MissingQ@ k -> 0, {n, 0, Max@ s}]]@ Table[Count[Range[10 k, 10 k + 100], ?PrimeQ], {k, 0, 10^5}] (* _Michael De Vlieger, Jul 12 2017; program writes "-1" for a(23) and a(24). *)

a(n) = my(k=0); while(1, if(primepi(10*k+100)-primepi(10*k)==n, return(k)); k++) \\ Felix Fröhlich, Jul 12 2017

a(n)=my(k); while(sum(p=10*k+1,10*k+99,isprime(p))!=n, k++); k \\ Charles R Greathouse IV, Jul 12 2017

Name clarified by FUNG Cheok Yin, Jul 12 2017

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A186405 Numbers k such that there are 13 primes between 100*k and 100*k + 99.

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A186406 Numbers k such that there are 14 primes between 100*k and 100*k + 99.

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A216289 Smallest k in which there are exactly n primes between 10*k and 10*k+9.

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A279862 a(n) = the smallest integer k where there are exactly n primes between 10k and 10k+100.

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A186405 Numbers k such that there are 13 primes between 100k and 100k + 99.

A186406 Numbers k such that there are 14 primes between 100k and 100k + 99.

A216289 Smallest k in which there are exactly n primes between 10k and 10k+9.