A186509 -id:A186509 - cp's OEIS frontend

A038822 Number of primes between 100n and 100n+99.

25, 21, 16, 16, 17, 14, 16, 14, 15, 14, 16, 12, 15, 11, 17, 12, 15, 12, 12, 13, 14, 10, 15, 15, 10, 11, 15, 14, 12, 11, 12, 10, 11, 15, 11, 14, 13, 12, 11, 11, 15, 9, 16, 9, 11, 12, 12, 12, 8, 15, 12, 11, 10, 10, 13, 13, 12, 10, 16, 7, 12, 11, 13, 15, 8, 11, 10, 12, 12, 13, 9, 10

Offset: 0

Jeff Burch

nonn

The number k first occurs in century A186311(k).

			a(3) = 16 because there are 16 primes between 300 and 399 (namely, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397).
a(4) = 17 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, 499).

George P. Loweke, The Lore of Prime Numbers. New York: Vantage Press (1982): 91.

T. D. Noe, Table of n, a(n) for n = 0..9999

Cf. A028505.
Cf. A181098 (centuries without primes).
Cf. A186393-A186408 (centuries having 1 to 16 primes), A186509 (17 primes).
Cf. A094892, A098592.

with(numtheory); A038822 := n->pi(100*n+99)-pi(100*n); seq(A038822(k), k=0..100); # Wesley Ivan Hurt, Oct 03 2013

Table[PrimePi[100n + 99] - PrimePi[100n], {n, 0, 71}]
Differences[PrimePi[100 Range[0,100]]] (* Harvey P. Dale, Feb 18 2021 *)

a(n)=sum(i=100*n,100*n+99,isprime(i)) \\ Charles R Greathouse IV, Apr 28 2015

a(n)= my(r=0, p=100*n, q=p+99); while((p=nextprime(p+1))<=q, r+=isprime(p)); r; \\ Ruud H.G. van Tol, Nov 17 2024

a(n) = pi(100n+99) - pi(100n). - Wesley Ivan Hurt, Oct 03 2013

Edited, corrected and extended by Robert G. Wilson v, Jan 29 2003

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

Original entry on oeis.org

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

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

A186394 Numbers k such that there are 2 primes between 100k and 100k + 99.

Original entry on oeis.org

3020, 3709, 4484, 4617, 4806, 4921, 5072, 5423, 5616, 6041, 6194, 6231, 6452, 6485, 6683, 6828, 7101, 7365, 7454, 7532, 7839, 8096, 8157, 8728, 8738, 9221, 9486, 9635, 9796, 10152, 10506, 10720, 10852, 11261, 11621, 11736, 11953, 11992

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 780 possible prime patterns for centuries having 2 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			3020 is in this sequence because there are 2 primes between 302000 and 302099 (302009 and 302053).

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))==2, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

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

def is_A186394(n):
    np0 = next_prime(next_prime(100*n))
    np1 = next_prime(np0)
    return np0 <= 100*n+99 and np1 > 100*n+99  # D. S. McNeil, Feb 21 2011

A186407 Numbers k such that there are 15 primes between 100k and 100k + 99.

Original entry on oeis.org

8, 12, 16, 22, 23, 26, 33, 40, 49, 63, 75, 94, 375, 424, 1131, 1572, 3442, 3922, 7393, 9780, 13939, 16528, 17492, 29673, 71338, 75877, 237421, 464977, 514483, 687352, 747574, 981953, 1040840, 1269778, 1298137, 1346413, 1790287, 1884223, 2330647, 2527249, 2601874, 2813749

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 12815608 possible prime patterns for centuries having 15 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			8 is in this sequence because there are 15 primes between 800 and 899 (809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883 and 887).

Brian Kehrig, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)

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))==15, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 4e9, if(p>N, if(s==15, print1((N\100)-1,", ")); s=1; N=100*(p\100+1),s++)) \\ Charles R Greathouse IV, Feb 21 2011

a(19)-a(42) from Charles R Greathouse IV, Feb 21 2011

A261571 Number of possible prime patterns for centuries having exactly n primes.

Original entry on oeis.org

1, 40, 780, 7528, 47878, 225044, 830270, 2459376, 5900602, 11555200, 18634704, 24942742, 27836859, 25913910, 20053913, 12815608, 6699888, 2829786, 948729, 245756, 47150, 6276, 518, 20

Offset: 0

Tim Johannes Ohrtmann, Aug 27 2015

The index of the final term is A364678(100) = 23. - Peter Munn, Sep 04 2023

Cf. A010956.
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. A186311 (first occurrences).
Cf. A364678.

A186395 Numbers k such that there are 3 primes between 100k and 100k + 99.

Original entry on oeis.org

588, 695, 797, 1430, 1621, 1751, 1809, 1869, 1904, 1913, 2042, 2067, 2123, 2127, 2322, 2471, 2505, 2562, 2734, 2833, 2862, 2874, 2935, 3023, 3077, 3134, 3371, 3380, 3552, 3611, 3679, 3703, 3707, 3725, 3878, 4046, 4167, 4215

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 7528 possible prime patterns for centuries having 3 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			588 is in this sequence because there are 3 primes between 58800 and 58899 (58831, 58889 and 58897).

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))==3, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 1e6, if(p>N, if(s==3, print1((N\100)-1, ", ")); s=1; N=100*(p\100+1), s++)) \\ Charles R Greathouse IV, Feb 21 2011

cp's OEIS Frontend

A038822 Number of primes between 100n and 100n+99.

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

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A186394 Numbers k such that there are 2 primes between 100*k and 100*k + 99.

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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.

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

A186394 Numbers k such that there are 2 primes between 100k and 100k + 99.

A186407 Numbers k such that there are 15 primes between 100k and 100k + 99.

A186395 Numbers k such that there are 3 primes between 100k and 100k + 99.