A038822 -id:A038822 - cp's OEIS frontend

A342839 Numbers k such that there are more primes in the interval [4k+1, 5k] than there are in the interval [3k+1, 4k].

1, 4, 7, 9, 10, 15, 16, 22, 23, 24, 25, 34, 36, 37, 39, 40, 47, 55, 56, 57, 58, 64, 67, 82, 84, 86, 87, 88, 91, 93, 94, 95, 96, 97, 98, 99, 100, 102, 104, 105, 106, 107, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 144, 146, 147, 148, 149, 150, 153

Offset: 1

Jon E. Schoenfield, Mar 23 2021

nonn

After a(876) = 11895, there are no more terms < 100000.
Conjecture: a(876) = 11895 is the final term.
There exist eight terms k for which A342068(k) != 5: A342068(k) = 2 for k = 1; A342068(k) = 3 for k = 47, 67, 95, and 1323; and A342068(k)=4 for k = 22, 57, and 102.

			The intervals [1, 100], [101, 200], [201, 300], [301, 400], and [401, 500] contain 25, 21, 16, 16, and 17 primes, respectively (cf. A038822); 17 > 16, so 100 is a term of the sequence.

Cf. A342068, A342069, A342070, A342071.

A164987 First pair of primes (p1, p2) that begin centuries of primes having the same prime configuration, ordered by increasing p2. Each configuration is allowed only once.

Original entry on oeis.org

390503, 480803, 351121, 566821, 78901, 578701, 323623, 606223, 326701, 645901, 619471, 745471, 655717, 842617, 437321, 855821, 854713, 876913, 811337, 915437, 561409, 920509, 515401, 956401, 452401, 1023601, 805633, 1049333, 247141, 1092541, 1037903, 1127603

Offset: 1

Ki Punches, Sep 03 2009 through Dec 06 2009

Rearranged the pairs of numbers so that the sequence of values of p2 increases. The first pair is for the primes 390500 + {3, 27, 39, 53, 81} and 480803 + {3, 27, 39, 53, 81}. There is a large, but finite number of terms. How many terms are there? - T. D. Noe, Feb 10 2013
The sequence lists the small prime twin centuries. As exploration goes into higher primes many are found to be triples or even higher. Example: 1072009 is a twin with 5179509, a triple with 7183109, quadruple with 8284709, quintuple with 8462609, and sextuple with 9739309, and there could be infinitely more. - Ki Punches, Dec 17 2009
The first two centuries without any primes start with 1671800 and 2637800. These are not included in the sequence since they do not have a first prime. However, if they were to be included they would be the 136th pair. - Andrew Howroyd, Feb 25 2018

			The primes in 480800..480899 are 480803, 480827, 480839, 480853, 480881 ending with 03, 27, 39, 53, 81. The primes in 390500..390599 end with the same digits, and no earlier pair has this quality. Hence a(1) = 390503 and a(2) = 480803.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A038822 (number of primes between 100n and 100n+99).

Cf. A181098, A186393-A186408 (centuries having 0 to 16 primes).

pSig[n_] := Prime[Range[PrimePi[100 n] + 1, PrimePi[100 (n + 1)]]] - 100 n; t = {}; c = {}; found = {}; Do[s = pSig[n]; If[Length[s] > 0 && ! MemberQ[found, s] && MemberQ[c, s], d = Mod[s[[1]], 100]; AppendTo[found, s]; AppendTo[t, {Position[c, s][[1, 1]]*100 + d, n*100 + d}]]; AppendTo[c, s], {n, 11000}]; Flatten[t] (* T. D. Noe, Feb 10 2013 *)

sig(c)={my(s=0); for(v=0,49,if(isprime(100*c+2*v+1),s+=2^v)); s}
pairs(n)={my(L=List(),M=Map(),c=0); while(#L<2*n, c++; my(s=sig(c),f=0); if(mapisdefined(M,s,&f), if(f&&s,my(d=2*valuation(s,2)+1); listput(L,100*f+d); listput(L,100*c+d); mapput(M,s,0)), mapput(M,s,c))); Vec(L)}
pairs(20) \\ Andrew Howroyd, Feb 25 2018

Terms rearranged by T. D. Noe, Feb 10 2013

A116356 Number of primes less than n*10^21.

Original entry on oeis.org

21127269486018731928, 41644391885053857293, 61943374158983520871, 82103246362658124007, 102160925813497229402, 122137912741771709423, 142048291427909819758, 161902001837504830333, 181706431926947074426, 201467286689315906290

Offset: 1

Robert G. Wilson v, Feb 04 2006

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..53
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]

Cf. A000720, A038801, A028505, A038812, A038813, A038814, A038815, A038816, A038817, A038818, A038819.

Cf. A038820, A038821, A038822, A080123, A080124, A080125, A080126, A080127, A080128, A080129.

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.

A160709 Centuries containing a prime number of primes.

Original entry on oeis.org

5, 14, 15, 20, 26, 30, 33, 35, 37, 39, 40, 45, 52, 55, 56, 60, 62, 63, 66, 70, 73, 75, 81, 87, 88, 89, 91, 93, 94, 96, 97, 98, 101, 108, 112, 115, 120, 122, 125, 131, 135, 140, 143, 155, 157, 166, 167, 168, 171, 175, 182, 183, 184, 185, 188, 189, 191, 192, 193, 196

Offset: 1

Jonathan Vos Post, May 25 2009

			a(1) = 5 because 5 is the first Century to have a prime number (17) of primes.

H. v. Eitzen, Table of n, a(n) for n = 1..10000

Cf. A000040, A038822.

A038822 := proc(n) numtheory[pi](n*100)-numtheory[pi]((n-1)*100) ; end: A160709 := proc(n) option remember ; local a; if n = 1 then 5 ; else for a from procname(n-1)+1 do if isprime( A038822(a) ) then RETURN(a) ; fi; od: fi; end: seq(A160709(n),n=1...90) ; # R. J. Mathar, May 27 2009

Select[Range[200],PrimeQ[PrimePi[100#]-PrimePi[100(#-1)]]&] (* Harvey P. Dale, Oct 03 2011 *)

{n: A038822(n) is in A000040}.

Extended by Ray Chandler, May 25 2009

More terms from R. J. Mathar, May 27 2009

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

A186396 Numbers k such that there are 4 primes between 100k and 100k + 99.

Original entry on oeis.org

314, 356, 524, 662, 831, 881, 1037, 1101, 1124, 1307, 1370, 1433, 1623, 1713, 1733, 1755, 1801, 1808, 1831, 1880, 1956, 2031, 2150, 2178, 2202, 2222, 2231, 2330, 2374, 2502, 2503, 2532, 2545, 2611, 2618, 2656, 2659, 2665

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 47878 possible prime patterns for centuries having 4 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			314 is in this sequence because there are 4 primes between 31400 and 31499 (31469, 31477, 31481 and 31489).

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))==4, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 1e6, if(p>N, if(s==4, print1((N\100)-1, ", ")); s=1; N=100*(p\100+1), s++)) \\ Charles R Greathouse IV, Feb 21 2011

A186397 Numbers k such that there are 5 primes between 100k and 100k + 99.

Original entry on oeis.org

188, 273, 377, 403, 438, 506, 598, 605, 732, 758, 790, 800, 866, 885, 916, 936, 972, 981, 1031, 1032, 1060, 1074, 1075, 1086, 1103, 1128, 1136, 1193, 1194, 1204, 1218, 1240, 1248, 1265, 1280, 1287, 1293, 1298, 1390, 1400

Offset: 1

Tim Johannes Ohrtmann, Feb 20 2011

nonn

There are 225044 possible prime patterns for centuries having 5 primes. - Tim Johannes Ohrtmann, Aug 27 2015

			188 is in this sequence because there are 5 primes between 18800 and 18899 (18803, 18839, 18859, 18869 and 18899).

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))==5, print1(n", "))); \\ Charles R Greathouse IV, Feb 21 2011

N=100; s=0; forprime(p=2, 1e6, if(p>N, if(s==5, print1((N\100)-1, ", ")); s=1; N=100*(p\100+1), s++)) \\ Charles R Greathouse IV, Feb 21 2011

cp's OEIS Frontend

A342839 Numbers k such that there are more primes in the interval [4*k+1, 5*k] than there are in the interval [3*k+1, 4*k].

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A164987 First pair of primes (p1, p2) that begin centuries of primes having the same prime configuration, ordered by increasing p2. Each configuration is allowed only once.

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A116356 Number of primes less than n*10^21.

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

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

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

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A160709 Centuries containing a prime number of primes.

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

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A186396 Numbers k such that there are 4 primes between 100*k and 100*k + 99.

A342839 Numbers k such that there are more primes in the interval [4k+1, 5k] than there are in the interval [3k+1, 4k].

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.

A186396 Numbers k such that there are 4 primes between 100k and 100k + 99.

A186397 Numbers k such that there are 5 primes between 100k and 100k + 99.