A095389 -id:A095389 - cp's OEIS frontend

A095390 Least inverse of A095389.

64, 28, 11, 14, 5, 3, 2, 1, 968

Offset: 0

Labos Elemer, Jun 16 2004

It is believed (or proved) that values at n > 8 indices do not occur, except once, at the beginning when 13 lesser twins in 210.0+R range, where R is the reduced residues modulo 210.

			a(13) = 0 because A095389(0) = 13.
a(1) = 28 means that in reduced residue system 210.28+R exactly 1 lesser-twin-prime arises; see A095389.
a(8) = 968 means that at surprisingly high density of twin primes [8 cases] occur in range of {210.968+r, 210.968+r+2}, as follows: {203309, 203321, 203339, 203351, 203381, 203417, 203429, 203459}.

Cf. A095389, A078859.

{k=0, ta=Table[0, {100000}]}; Do[{m=0};Do[s=210k+r; s1=210k+r+2; If[PrimeQ[s]&&PrimeQ[s+2], m=m+1], {r, 1, 210}];ta[[k]]=m, {k, 1, 100000}]; Table[Min[Flatten[Position[ta, j]]], {j, 0, 15}]

a(n) = min{x: A095389(x) = n}

Edited by Charles R Greathouse IV, Oct 27 2010

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

A095391 a(n) is the least x such that A094892(x)=n.

Original entry on oeis.org

1751793, 235449, 60110, 10471, 17110, 8495, 6288, 3182, 2452, 1349, 331, 348, 446, 223, 249, 205, 111, 67, 55, 63, 28, 37, 14, 21, 18, 11, 10, 6, 551, 5, 4, 7, 3, 2

Offset: 0

Labos Elemer, Jun 16 2004

			a[0]=1751793 because there are no primes between 210*1751793 and 210*1751794.
a[1]=235449 because there is one prime between 210*235449 and 210*235450.

Cf. A078859, A095389, A008364.

ta=Table[0, {up}]; Do[{m=0};Do[s=210*k+r; s1=210*k+r+2; If[PrimeQ[s], m=m+1], {r, 1, 210}];ta[[k]]=m, {k, 1, up}] Table[Min[Flatten[Position[ta, j]]], {j, 1, 48}]

Edited by Don Reble, Jun 16 2004

A095392 Numbers n such that more than half of the reduced-residue system modulo 210 consists of primes in the following sense: in {210n + R} more than 24 = phi(210)/2 primes occur, i.e., 25-33, 35, 46.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 20, 22, 23, 26, 27, 29, 31, 36, 41, 44, 46, 48, 56, 59, 70, 72, 74, 95, 109, 113, 114, 127, 132, 136, 148, 312, 321, 347, 428, 506, 538, 551, 1274, 1296, 1442, 2875, 4576, 5504, 6928, 7870, 12880, 15745, 17518

Offset: 1

Labos Elemer, Jun 16 2004

nonn

			210n + r, where r runs through RRS of 210 corresponds to prime-difference patterns with several relatively small first prime differences.
n=18543: 210*18543 + r includes 26 primes with the following difference pattern: {2,4,2,4,30,18,2,10,6,12,2,18,6,10,2,12,12,4,6,8,6,6,4,2,10}.

Cf. A095389, A095390, A095391.

{k=0};Do[{m=0}; Do[s=210k+r; s1=210k+r+2;If[PrimeQ[s], m=m+1], {r, 1, 210}]; If[Greater[m, 24], Print[{m, k}]], {k, 0, 10000000}]

Solutions to A095390(x) > 24 = phi(210).

A095393 Terms n are such that exactly half[=24] of the {210n+r} set is prime. Here r runs through the reduced residue system mod 210 (RRS[210]).

Original entry on oeis.org

18, 19, 25, 33, 39, 42, 61, 65, 85, 86, 92, 100, 102, 112, 154, 175, 203, 259, 265, 281, 369, 380, 384, 441, 495, 518, 611, 649, 748, 840, 1083, 1355, 1376, 1515, 1559, 1610, 1703, 1874, 2226, 2355, 2464, 2667, 2716, 3371, 3577, 4011, 4021, 4791, 5290, 5808

Offset: 1

Labos Elemer, Jun 16 2004

nonn

			For n=92269 the 24 primes are:
{19376491,19376501,19376503,19376507,19376521,19376527,19376543,19376563,
19376569,19376573,19376579,19376597,19376629,19376633,19376639,19376647,
19376653,19376657,19376663,19376671,19376677,19376683,19376689,19376699}

Cf. A095389-A095392.

{k=0, u=0, ta=Table[0, {256}]}; Do[{m=0};w=k;Do[s=210k+r; s1=210k+r+2; If[PrimeQ[s], m=m+1], {r, 1, 210}]; If[Equal[m, 24], Print[k];ta[[u]]=k;u=u+1], {k, 0, 1000000}]

In 210n+RRS[210] the number of primes is 24=phi[210]/2.

cp's OEIS Frontend

A095390 Least inverse of A095389.

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

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A095391 a(n) is the least x such that A094892(x)=n.

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A095392 Numbers n such that more than half of the reduced-residue system modulo 210 consists of primes in the following sense: in {210n + R} more than 24 = phi(210)/2 primes occur, i.e., 25-33, 35, 46.

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A095393 Terms n are such that exactly half[=24] of the {210n+r} set is prime. Here r runs through the reduced residue system mod 210 (RRS[210]).

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