A038822 -id:A038822 - cp's OEIS frontend

A181098 Primefree centuries (i.e., numbers k such that no prime exists between 100k and 100k+99).

16718, 26378, 31173, 39336, 46406, 46524, 51782, 55187, 58374, 58452, 60129, 60850, 63338, 63762, 67898, 69587, 71299, 75652, 78035, 78269, 80277, 83674, 84213, 89052, 95490, 97080, 100881, 101527, 103438, 105916, 111772, 112967

Offset: 1

Jeff Burch, Oct 02 2010

nonn

The first consecutive terms are 473267, 473268; see A190639. - M. F. Hasler, May 15 2011

			16718 is a term because there is no prime between 1671800 and 1671899.

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. A186393-A186408 (1 to 16 primes), A186509 (17 primes), A361723 (18 primes).

Flatten[Position[Differences[PrimePi[100*Range[0,113000]]],0]]-1 (* Harvey P. Dale, Dec 18 2021 *)

is(n)=nextprime(100*n)>100*n+99 \\ Charles R Greathouse IV, Apr 28 2015

a(n) = n + 100n/log n - O(n/log^2 n). - Charles R Greathouse IV, Sep 08 2017

A186311 Least k such that the interval 100k to 100k+99 has exactly n primes.

Original entry on oeis.org

16718, 1559, 3020, 588, 314, 188, 186, 59, 48, 41, 21, 13, 11, 19, 5, 8, 2, 4, 1228537713709, 14688670051164208, 203860951641372730864, 1

Offset: 0

T. D. Noe, Feb 22 2011

It is known that a(25)=0. Terms for n = 22 and 23 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
To see that Dickson's conjecture is applicable to the preceding statement, the appropriate general sequence to consult is A364678, which affirms that 23 primes are permissible between adjacent multiples of 100, as opposed to in an arbitrary interval of 99 integers. - Peter Munn, Sep 04 2023
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

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

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), 155-161.

Cf. A038822 (number of primes between 100n and 100n+99).
Cf. A181098 (centuries without primes).
Cf. A186393-A186408 (centuries having 1 to 16 primes).
Cf. A186509 (centuries having 17 primes).
Cf. A361723 (centuries having 18 primes).
Cf. A020497, A261571, A364678.

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

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

a(18) from Donovan Johnson, Feb 28 2011
a(19) from Brian Kehrig, Apr 08 2023
a(20)-a(21) from Brian Kehrig, May 28 2024

A186393 Numbers k such that there is 1 prime between 100k and 100k + 99.

Original entry on oeis.org

1559, 2683, 4133, 10048, 11400, 12727, 12800, 13572, 14223, 14443, 14514, 14680, 14913, 15536, 15619, 16538, 16557, 17334, 19043, 20452, 20465, 20522, 21162, 21663, 22440, 22832, 23055, 23144, 23214, 23460, 24833, 25139, 25278, 25980, 26207, 26257, 26702, 26747, 27536, 27878, 28448, 28671, 29180, 29873, 30212, 30232

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 40 possible prime patterns for centuries having 1 prime. - Tim Johannes Ohrtmann, Aug 27 2015

			1559 is in this sequence because there is 1 prime between 155900 and 155999 (155921).

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), A186394-A186408 (2 to 16 primes), A186509 (17 primes), A361723 (18 primes).

Select[Range[31000],PrimePi[100 #+99]-PrimePi[100 #]==1&] (* Harvey P. Dale, Dec 16 2011 *)

isA186393(n)=my(k=nextprime(100*n)); n=100*(n+1); kn \\ Charles R Greathouse IV, Feb 21 2011

a(4)-a(46) from Charles R Greathouse IV, Feb 21 2011

A186408 Numbers k such that there are 16 primes between 100k and 100k + 99.

Original entry on oeis.org

2, 3, 6, 10, 42, 58, 194, 230, 12463, 8392963, 24662691, 37400476, 163061323, 205481131, 278399797, 313114319, 481863166, 494959102, 656914015, 776749247, 960655996, 980373049, 1097546872, 1156724143, 2013136112, 2245034146, 3416649829, 3606810631, 4141180699, 5928231877, 6569717174, 6594050440, 7240502155, 7492029097, 8995086259

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 6699888 possible prime patterns for centuries having 16 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			2 is in this sequence because there are 16 primes between 200 and 299 (211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283 and 293).

Brian Kehrig, 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-A186407 (1 to 15 primes), A186509 (17 primes), A361723 (18 primes).

for(n=1, 1e6, if(sum(k=100*n,100*(n+1), ispseudoprime(k))==16, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 4e9, if(p>N, if(s==16, print1((N\100)-1,", ")); s=1; N=100*(p\100+1),s++)) \\ Charles R Greathouse IV, Feb 21 2011

a(9)-a(12) from Charles R Greathouse IV, Feb 21 2011

a(13)-a(35) from T. D. Noe, Feb 23 2011

A186509 Numbers k such that there are 17 primes between 100k and 100k + 99.

Original entry on oeis.org

4, 14, 7837, 27049, 144997771, 651186838, 12779564974, 22369949923, 149621468452, 225012717952, 240728320642, 586832463472, 766964610742, 939742446571, 949543082647, 1908189311558, 2693729868901, 2701032171244, 3465208973035, 3489960850720, 3910908182851

Offset: 1

T. D. Noe, Feb 22 2011

nonn

There are 2829786 possible prime patterns for centuries having 17 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			4 is in this sequence because there are 17 primes between 400 and 499 (401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491 and 499).

Brian Kehrig, Table of n, a(n) for n = 1..365

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

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

a(7)-a(15) from Donovan Johnson, Feb 28 2011

Terms a(16) and beyond from Brian Kehrig, Mar 21 2023

A361723 Numbers k such that there are 18 primes between 100k and 100k + 99.

Original entry on oeis.org

1228537713709, 23352869714018, 28703237474266, 144785865481702, 161394923966449, 168975708209638, 174748809066898, 207552241231357, 278215179205531, 312303328909720, 592248982143877, 812939886634531, 939100782752014, 983930290209021, 1111161494544274

Offset: 1

Brian Kehrig, Mar 21 2023

nonn

There are A261571(18) = 948729 possible patterns for centuries having 18 primes.

			1228537713709 is in the sequence because there are 18 primes between 122853771370900 and 122853771370999: 122853771370900 + x, where x is one of (1, 3, 7, 19, 21, 27, 31, 33, 37, 49, 51, 61, 69, 73, 87, 91, 97, or 99).

Brian Kehrig, Table of n, a(n) for n = 1..39 (terms up to 10^16)
Note (Apr 24 2024): An older version of the b-file missed a(33) and a(38). The present b-file is correct.

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).
Cf. A261571 (number of patterns for centuries with n primes).

isok(k) = sum(i=0, 99, isprime(100*k + i)) == 18; \\ Michel Marcus, Mar 23 2023

A098592 Number of primes between n30 and (n+1)30.

Original entry on oeis.org

10, 7, 7, 6, 5, 6, 5, 6, 5, 5, 4, 6, 5, 4, 6, 5, 5, 2, 5, 5, 5, 6, 4, 4, 4, 5, 3, 6, 4, 4, 4, 4, 4, 5, 5, 4, 6, 3, 3, 4, 5, 4, 4, 6, 2, 3, 3, 5, 4, 7, 2, 5, 4, 6, 3, 4, 4, 3, 4, 4, 3, 2, 7, 3, 3, 3, 5, 5, 3, 5, 3, 5, 2, 3, 4, 4, 5, 3, 4, 7, 3, 4, 3, 1, 5, 3, 3, 3, 4, 7, 5, 4, 3, 5, 3, 4, 4, 3, 4, 2, 4, 3, 5, 2, 2, 3

Offset: 0

Hugo Pfoertner, Sep 16 2004

Number of nonzero bits in A098591(n).
The number a(n) is < 8 except for n=0. - Pierre CAMI, Jun 02 2009
For references to positions where a(n) = 7 and related explanation, see A100418. - Peter Munn, Sep 06 2023

			a(1)=7 because there are 7 primes in the interval (30,60): 31,37,41,43,47,53,59.
a(26)=3 because the interval of length 30 following 26*30=780 contains 3 primes: 787, 797 and 809.

Dennis Martin, Proofs Regarding Primorial Patterns [Cached copy, with permission of the author].
Hugo Pfoertner, Patterns count table.

Cf. A000040 (prime numbers), A098591 (packed representation of the primes mod 30), A100418, A185641.

Cf. A038822, A094892.

FORTRAN
```
! See links given in A098591.
```

a(n) = primepi(30*(n+1)) - primepi(30*n); \\ Michel Marcus, Apr 04 2020

from sympy import primerange
def a(n): return len(list(primerange(n*30, (n+1)*30)))
print([a(n) for n in range(106)]) # Michael S. Branicky, Oct 07 2021

Edited by N. J. A. Sloane, Jun 12 2009 at the suggestion of R. J. Mathar

A094892 a(n) is the number of primes between n210 and (n+1)210.

Original entry on oeis.org

46, 35, 33, 32, 30, 29, 27, 31, 27, 27, 26, 25, 30, 26, 22, 27, 26, 27, 24, 24, 26, 23, 26, 26, 22, 24, 26, 27, 20, 25, 23, 25, 23, 24, 22, 23, 26, 21, 21, 24, 21, 26, 24, 23, 25, 22, 25, 20, 25, 22, 21, 22, 21, 22, 21, 18, 26, 22, 21, 26, 23, 24, 22, 19, 21, 24, 21, 17, 23

Offset: 0

Labos Elemer, Jun 16 2004

nonn

Arbitrarily long subsequences of consecutive 0's occur. a(n) is always <= 46. All values below 34 occur (see A095391); does 34?

			a(0) = 46 because there are 46 primes between 0*210 and 1*210.
a(1) = 35 because there are 35 primes between 1*210 and 2*210.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A008364, A038822, A078859, A095389, A098592.

[46] cat [#PrimesInInterval(210*n, 210*(n+1)): n in [1..80]]; // Vincenzo Librandi, Jul 08 2018

a[n_]:=PrimePi[210 (n + 1)] - PrimePi[210 n]; Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, Jul 08 2018 *)

a(n) = primepi(210*(n+1)) - primepi(210*n); \\ Ruud H.G. van Tol, Oct 27 2024

a(n) = my(res = 0); forprime(p = n*210, (n+1)*210, isprime(p) && res++); res \\ David A. Corneth and Ruud H.G. van Tol, Oct 27 2024

Edited by Don Reble, Jun 16 2004

Examples corrected by Matthew Vandermast, Jun 17 2004

A342070 Numbers k such that there are more primes in the interval [2k+1, 3k] than there are in the interval [k+1, 2*k].

Original entry on oeis.org

5, 8, 14, 18, 20, 29, 47, 48, 67, 68, 81, 95, 109, 110, 111, 113, 168, 173, 277, 278, 280, 281, 283, 284, 288, 293, 295, 296, 710, 711, 713, 1323

Offset: 1

Jon E. Schoenfield, Mar 23 2021

Conjecture: 1323 is the final term.

If there are at least as many primes in [1, m] as there are in [m+1, 2*m] for all positive integers m, then this sequence consists of the numbers k such that A342068(k)=3.

			The intervals [1, 100], [101, 200], and [201, 300] contain 25, 21, and 16 primes respectively (cf. A038822); 16 < 21, so 100 is not a term of the sequence.
The intervals [1, 20], [21, 40], and [41, 60] contain 8, 4, and 5 primes, respectively; 5 > 4, so 20 is a term.

Cf. A038822, A342068, A342069, A342071, A342839.

from sympy import primepi
def ok(n): return primepi(3*n) > 2*primepi(2*n) - primepi(n)
print([m for m in range(9999) if ok(m)]) # Michael S. Branicky, Mar 23 2021

A342071 Numbers k such that there are more primes in the interval [3k+1, 4k] than there are in the interval [2k+1, 3k].

Original entry on oeis.org

12, 19, 22, 32, 42, 45, 49, 50, 52, 54, 57, 59, 70, 71, 72, 73, 74, 75, 101, 102, 115, 116, 117, 121, 122, 123, 124, 126, 132, 143, 180, 182, 184, 185, 186, 187, 188, 189, 190, 192, 194, 195, 197, 268, 269, 309, 310, 311, 312, 322, 323, 325, 326, 327, 328, 329

Offset: 1

Jon E. Schoenfield, Mar 23 2021

nonn

After a(194)=3977, there are no more terms < 100000.
Conjecture: a(194)=3977 is the final term.
For each of the first 194 terms k, there are at least as many primes in [1, k] as there are in [k+1, 2*k], and at least as many primes in [k+1, 2*k] as there are in [2*k+1, 3*k], so A342068(k)=4.

			The intervals [1, 100], [101, 200], [201, 300], and [301, 400] contain 25, 21, 16, and 16 primes respectively (cf. A038822); the 4th interval does not contain more primes than does the 3rd, so 100 is not a term of the sequence.
However, the intervals [1, 101], [102, 202], [203, 303], and [304, 404] contain 26, 20, 16, and 17 primes, respectively; 17 > 16, so 101 is a term.

Cf. A342068, A342069, A342070, A342839.

cp's OEIS Frontend

A181098 Primefree centuries (i.e., numbers k such that no prime exists between 100*k and 100*k+99).

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A186311 Least k such that the interval 100k to 100k+99 has exactly n primes.

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A186393 Numbers k such that there is 1 prime between 100*k and 100*k + 99.

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A186408 Numbers k such that there are 16 primes between 100*k and 100*k + 99.

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A186509 Numbers k such that there are 17 primes between 100*k and 100*k + 99.

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A361723 Numbers k such that there are 18 primes between 100*k and 100*k + 99.

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A098592 Number of primes between n*30 and (n+1)*30.

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A181098 Primefree centuries (i.e., numbers k such that no prime exists between 100k and 100k+99).

A186393 Numbers k such that there is 1 prime between 100k and 100k + 99.

A186408 Numbers k such that there are 16 primes between 100k and 100k + 99.

A186509 Numbers k such that there are 17 primes between 100k and 100k + 99.

A361723 Numbers k such that there are 18 primes between 100k and 100k + 99.

A098592 Number of primes between n30 and (n+1)30.

A094892 a(n) is the number of primes between n210 and (n+1)210.

A342070 Numbers k such that there are more primes in the interval [2k+1, 3k] than there are in the interval [k+1, 2*k].

A342071 Numbers k such that there are more primes in the interval [3k+1, 4k] than there are in the interval [2k+1, 3k].