cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A078864 Smallest primes from A001359, each belonging to those different residue class of mod 210 which are listed in A078859. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 419, 1427
Offset: 1

Views

Author

Labos Elemer, Dec 13 2002

Keywords

Examples

			Several terms are equal to corresponding ones in A078859, while others are larger like: 1427=210.6+167 where r=167 is in A078859.

Crossrefs

Cf. A001359, A008364, A078859-A078864.

Programs

  • Mathematica
    f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 2]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 10000}]; t

A095389 a(n) is the number of residues from reduced residue system, R, modulo 210 such that both R and R+2 are primes, i.e., both 210n+r and 210n+r+2 are primes at fixed n.

Original entry on oeis.org

13, 7, 6, 5, 5, 4, 6, 5, 5, 6, 6, 2, 6, 2, 3, 6, 7, 3, 4, 6, 6, 4, 5, 4, 2, 3, 6, 4, 1, 4, 2, 5, 5, 3, 4, 4, 2, 2, 2, 4, 3, 2, 5, 2, 5, 2, 4, 4, 3, 5, 2, 2, 4, 2, 3, 2, 4, 4, 3, 1, 1, 4, 1, 2, 0, 6, 5, 2, 3, 4, 1, 0, 4, 1, 5, 1, 4, 3, 1, 3, 3, 3, 3, 3, 5, 7, 3, 2, 2, 0, 3, 3, 4, 2, 3, 4, 2, 4, 4, 3, 4, 2, 6, 3, 1
Offset: 0

Views

Author

Labos Elemer and Enoch Haga, Jun 15 2004

Keywords

Comments

Since arbitrarily large prime gaps occur, several consecutive zeros may arise in the sequence.

Examples

			n=0: only 13+2=15 integers correspond to the condition: {11,17,29,41,59,71,101,107,137,149,179,191,197}, so a[0]=13; see A078859.
n=11: only 2 twins were found, {2339,2341} and {2381,2383} corresponding to residue pairs {29,31} and {71,73}.

Links

Crossrefs

Cf. A001359, A078859.

Programs

  • Mathematica
    {k =0, ta=Table[0, {100}]}; 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, 100}];ta
(* Second program: *)
With[{P = Product[Prime@ i, {i, 4}]}, Function[R, Array[Count[R + P #, k_ /; Times @@ Boole@ PrimeQ@ {k, k + 2} == 1] &, 105, 0]]@ Select[Partition[Select[Range[P + 1], CoprimeQ[#, P] &], 2, 1], Differences@ # == {2} &][[All, 1]]] (* Michael De Vlieger, May 15 2017 *)

A094892 a(n) is the number of primes between n*210 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

Views

Author

Labos Elemer, Jun 16 2004

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Magma
    [46] cat [#PrimesInInterval(210*n, 210*(n+1)): n in [1..80]]; // Vincenzo Librandi, Jul 08 2018
  • Mathematica
    a[n_]:=PrimePi[210 (n + 1)] - PrimePi[210 n]; Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, Jul 08 2018 *)
  • PARI
    a(n) = primepi(210*(n+1)) - primepi(210*n); \\ Ruud H.G. van Tol, Oct 27 2024
  • PARI
    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

Extensions

Edited by Don Reble, Jun 16 2004
Examples corrected by Matthew Vandermast, Jun 17 2004

A078860 Least positive residues [mod 210] representing those residue classes which can be lesser of prime pairs from A029710.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 139, 163, 169, 187, 193
Offset: 1

Views

Author

Labos Elemer, Dec 13 2002

Keywords

Crossrefs

Cf. A029710, A008364, A078859.

Programs

  • Mathematica
    t=Flatten[Position[Table[GCD[w, 210], {w, 1, 210}], 1]] t2=Intersection[t, t+4]-4

Formula

Intersection[RRS(210), 4+RRS{210)]-4 and {7}. RRS[210]=reduced residue system of 210=first 48=phi[210] terms of A008364; additional term 7 is a singular cases; 210k+r generates complete A029710 with suitable k and r taken from these 15+1 numbers.

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

Views

Author

Labos Elemer, Jun 16 2004

Keywords

Examples

			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.

Crossrefs

Cf. A078859, A095389, A008364.

Programs

  • Mathematica
    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}]

Extensions

Edited by Don Reble, Jun 16 2004

A078861 Least positive residues [mod 210] representing those residue classes which can be smaller prime of a difference 6 taken from A031924.

Original entry on oeis.org

11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 121, 131, 137, 143, 151, 157, 163, 167, 173, 181, 187, 191, 193
Offset: 1

Views

Author

Labos Elemer, Dec 13 2002

Keywords

Crossrefs

Cf. A031924, A008364, A078859, A078860.

Programs

  • Mathematica
    t=Flatten[Position[Table[GCD[w, 210], {w, 1, 210}], 1]] t2=Intersection[t, t+6]-6

Formula

Intersection[RRS(210), 6+RRS{210)]-6. RRS[210]=reduced residue system of 210=first 48=phi[210] terms of A008364; 210k+r generates complete A031924 with suitable k and r taken from these 30 numbers.

A095390 Least inverse of A095389.

Original entry on oeis.org

64, 28, 11, 14, 5, 3, 2, 1, 968
Offset: 0

Views

Author

Labos Elemer, Jun 16 2004

Keywords

Comments

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.

Examples

			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}.

Crossrefs

Cf. A095389, A078859.

Programs

  • Mathematica
    {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}]

Formula

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

Extensions

Edited by Charles R Greathouse IV, Oct 27 2010

A078862 Smallest primes from A031924, each belonging to those different residue class of mod 210 which are listed in A078861. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

1901, 433, 647, 23, 31, 3607, 251, 47, 53, 61, 1117, 73, 83, 727, 941, 733, 947, 331, 131, 557, 353, 151, 157, 373, 167, 173, 601, 607, 3761, 1033
Offset: 1

Views

Author

Labos Elemer, Dec 13 2002

Keywords

Examples

			Several terms are equal to corresponding ones in A078861, while others are larger like: 1033=4.210+193, where r=193 is in A078861.

Crossrefs

Cf. A031924, A008364, A078861, A078859-A078864.

Programs

  • Mathematica
    f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 6]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 1000}]; t

A078863 Smallest primes from A029710, each belonging to those different residue class of mod 210 which are listed in A078860. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 349, 163, 379, 397, 193
Offset: 1

Views

Author

Labos Elemer, Dec 13 2002

Keywords

Examples

			Several terms are equal to corresponding ones in A078860, while others are larger like: 397=210.1+187 where r=187 is in A078860.

Crossrefs

Cf. A031924, A008364, A078860, A078859-A078864.

Programs

  • Mathematica
    f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 4]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 10000}]; t
Showing 1-9 of 9 results.

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