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.

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A210476 Let p_(4,3)(m) be the m-th prime == 3 (mod 4). Then a(n) is the smallest p_(4,3)(m) such that the interval(p_(4,3)(m)*n, p_(4,3)(m+1)*n) contains exactly one prime == 3(mod 4).

Original entry on oeis.org

7, 67, 43, 67, 67, 191, 883, 43, 643, 379, 739, 103, 463, 643, 487, 883, 1303, 3847, 1447, 13963, 1087, 8863, 1999, 8167, 7687, 8443, 2707, 2203, 11083, 3463, 7687, 31387, 8419, 15919, 12979, 10099, 26683, 22027, 46687, 79687, 15439, 65839, 46723, 44683, 14887, 58963, 13879, 26947, 77587
Offset: 2

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 23 2013

Keywords

Comments

The limit of a(n) as n goes to infinity is infinity.
Conjecture: every a(n), except for a(7) = 191, is the lesser of a pair of cousin primes p and p+4, (see A023200).

Crossrefs

Cf. A195325, A207820, A210465, A210467, A210475.

Programs

  • Mathematica
    myPrime=Select[Table[Prime[n],{n,3000000}],Mod[#,4]==3&];
binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]0&]]]+offset-1]];
z=1;(*contains exactly ONE myPrime in the interval*)
Table[myPrime[[NestWhile[#1+1&,1,!((nextMyPrime[n myPrime[[#1]],z+1]>n myPrime[[#1+1]]))&]]],{n,2,30}]

A218561 4-gap primes: Prime p is a term iff there is no prime between 4*p and 4*nextprime(p), where nextprime=A151800.

Original entry on oeis.org

29, 71, 137, 197, 239, 269, 347, 419, 431, 641, 659, 809, 821, 1061, 1091, 1151, 1289, 1489, 1607, 1721, 1783, 1877, 1949, 1993, 2083, 2141, 2267, 2339, 2381, 2389, 2549, 2729, 2801, 2833, 2969, 2999, 3019, 3041, 3217, 3253, 3299, 3329, 3389, 3461
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 02 2012

Keywords

Examples

			29 is in the sequence since there are no primes in the interval(4*29,4*31)=(116,124)

Links

Crossrefs

Cf. A195270, A195271, A195377, A195325, A195329, A195379.

Programs

A195465 The first a(n) n-gap primes are lessers of twin primes, a(n) maximal.

Original entry on oeis.org

0, 5, 5, 17, 5, 6, 14, 6, 24, 75, 2, 4, 27, 11, 48, 50, 46, 9, 21, 7, 16, 137, 4, 55, 85, 14, 111, 24, 102, 291, 67, 89, 155, 180, 137, 330, 127, 413, 250, 241, 332, 619, 139, 234, 453, 929, 94, 160, 169, 22, 131, 434
Offset: 1

Views

Author

Vladimir Shevelev, Sep 19 2011

Keywords

Comments

For definition of n-gap primes, see comment to A195270.
Conjecture: a(n)>0 for n>1. This conjecture is equivalent to the conjecture that all terms of A195325 are lessers of twin primes.

Crossrefs

Cf. A195325, A080192, A195270, A195271, A164368, A194658, A164294.

Programs

  • Maple
    a:= proc(n) local i, p, q;
      p, q:= 2, 3;
      for i from 0 do
        while nextprime(n*p) < (n*q) do
          p, q:= q, nextprime(q)
        od;
        if not isprime(p+2) then return i fi;
        p, q:= q, nextprime(q)
      od
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 20 2011
  • Mathematica
    a[n_] := a[n] = Module[{i, p = 2, q = 3}, For[i = 0, True, i++, While[NextPrime[n p] < n q, p = q; q = NextPrime[q]]; If[!PrimeQ[p+2], Return[i]]; p = q; q = NextPrime[q]]];
Array[a, 20] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)
Previous Showing 11-13 of 13 results.

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