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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A222219 Numbers n such that n and n + 18 are prime and there is a power of two in the interval (n,n+18).

Original entry on oeis.org

5, 11, 13, 19, 23, 29, 53, 61, 113, 239, 251, 503, 1013, 1021, 4093, 8191, 65519, 65521, 262133, 1048571, 4194301, 302231454903657293676533
Offset: 1

Views

Author

Brad Clardy, Feb 23 2013

Keywords

Comments

It is a conjecture that this sequence is finite. A search around 2^n was done up to 2^1500.

Crossrefs

Cf. A153418, A221211.

Programs

  • Magma
    //Program finds primes separated by an even number (called gap) which
//have a power of two between them. Program starts with the smallest
//power of two above gap. Primes less than this starting point can be
//checked by inspection.
gap:=18;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
    powerof2:=2^i;
    for k:=powerof2-gap+1 to powerof2-1 by 2 do
        if (IsPrime(k) and IsPrime(k+gap)) then k;
        end if;
    end for;
end for;
  • Mathematica
    Flatten[Table[Select[2^n-Range[17],AllTrue[{#,#+18},PrimeQ]&],{n,4,80}]]// Sort (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 04 2019 *)

A106065 Primes p such that 1*p+18 and 18*p+1 are primes.

Original entry on oeis.org

11, 29, 41, 71, 79, 139, 149, 181, 251, 379, 401, 431, 491, 569, 659, 701, 709, 809, 821, 991, 1021, 1051, 1231, 1289, 1381, 1471, 1549, 1759, 1871, 1931, 1999, 2069, 2221, 2251, 2381, 2459, 2591, 2671, 2711, 2909, 2939, 3301, 3371, 3499, 3511, 3539, 3709
Offset: 1

Views

Author

Zak Seidov, May 07 2005

Keywords

Comments

Subsequence of A153418. - Michel Marcus, Dec 27 2014

Crossrefs

Cf. A153418 (p and p+18 are prime).

Programs

  • Magma
    [p: p in PrimesUpTo(10000)| IsPrime(p+18) and IsPrime(18*p+1)] // Vincenzo Librandi, Nov 13 2010
  • Mathematica
    Select[Prime[Range[220]], PrimeQ[18#+1]&&PrimeQ[1#+18]&]

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109
Offset: 1

Views

Author

Felix Fröhlich, Jul 28 2019

Keywords

Comments

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

Examples

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Links

Crossrefs

Cf. A231608.
Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

Programs

  • PARI
    row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array
Previous Showing 11-13 of 13 results.

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