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 1-6 of 6 results.

A158714 Primes p such that p1 = ceiling(p/2) + p is prime and p2 = floor(p1/2) + p1 is prime.

Original entry on oeis.org

3, 19, 67, 307, 379, 467, 547, 587, 739, 859, 1259, 1699, 1747, 1867, 2027, 2699, 2819, 3259, 3539, 4019, 4507, 5059, 5779, 7547, 8219, 8539, 8747, 8819, 9547, 10067, 10499, 10667, 11939, 13259, 13627, 13859, 14939, 17659, 17707, 17987, 18859
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Comments

All a(n) == 3 (mod 8), as this is necessary for p, p1 and p2 to be odd. - Robert Israel, May 11 2014

Examples

			67 is in the sequence because 67, ceiling(67/2) + 67 = 101 and floor(101/2) + 101 = 151 are all primes.

Links

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A242366.

Programs

  • Maple
    N:= 10^5; # to get all entries <= N
filter:= proc(p)
     local p1,p2;
     if not isprime(p) then return false fi;
     p1:= ceil(p/2)+p;
     if not isprime(p1) then return false fi;
     p2:= floor(p1/2)+p1;
     isprime(p2);
end proc;
select(filter,[seq(2*i+1,i=1..floor((N-1)/2)]; # Robert Israel, May 09 2014
  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p=Ceiling[p/2]+p],If[PrimeQ[p=Floor[p/2]+p],AppendTo[lst,Prime[n]]]],{n,7!}];lst

Extensions

Definition corrected by Robert Israel, May 09 2014

A158719 Primes p such that p1 = floor(p/2)+p is not prime and p2 = ceiling(p/2)+p is not prime, p3 = floor(p1/2)+p1 is not prime and p5 = ceiling(p1/2)+p1 is not prime, p4 = floor(p2/2)+p2 is not prime and p6 = ceiling(p2/2)+p2 is not prime.

Original entry on oeis.org

83, 97, 113, 227, 229, 251, 269, 271, 277, 283, 313, 317, 331, 353, 389, 397, 419, 433, 457, 463, 491, 503, 509, 523, 557, 563, 593, 599, 601, 617, 641, 653, 683, 691, 733, 743, 751, 757, 761, 773, 797, 823, 829, 857, 863, 937, 941, 971, 977, 1013, 1031, 1049
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[ !PrimeQ[p1=Floor[p/2]+p]&&!PrimeQ[p2=Ceiling[p/2]+p],If[ !PrimeQ[p3=Floor[p1/2]+p1]&&!PrimeQ[p5=Ceiling[p1/2]+p1],If[ !PrimeQ[p4=Floor[p2/2]+p2]&&!PrimeQ[p6=Ceiling[p2/2]+p2],AppendTo[lst,Prime[n]]]]],{n,6!}];lst
nonpQ[p_]:=Module[{p1=Floor[p/2]+p,p2=Ceiling[p/2]+p},NoneTrue[ {p1,p2,Floor[ p1/2]+p1,Ceiling[p1/2]+p1,Floor[p2/2]+p2,Ceiling[p2/2]+ p2},PrimeQ]]; Select[Prime[Range[200]],nonpQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 21 2019 *)

A158720 Primes p such that Floor[p/3]+p is prime.

Original entry on oeis.org

2, 13, 31, 67, 73, 103, 181, 193, 211, 307, 337, 433, 463, 571, 577, 607, 643, 661, 733, 757, 787, 823, 937, 967, 991, 1021, 1117, 1201, 1291, 1567, 1597, 1621, 1723, 1783, 1831, 1993, 2017, 2083, 2143, 2251, 2281, 2287, 2341, 2377, 2521, 2593, 2647, 2713
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Comments

Floor[13/3]+13=17, ...

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714, A158719

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[Floor[p/3]+p],AppendTo[lst,p]],{n,6!}];lst

A158721 Primes p such that (p + 1)/3 + p is prime.

Original entry on oeis.org

2, 5, 17, 23, 53, 59, 113, 149, 167, 179, 197, 233, 269, 347, 359, 449, 557, 563, 617, 647, 683, 743, 773, 797, 827, 863, 977, 1049, 1103, 1187, 1319, 1367, 1373, 1409, 1499, 1583, 1607, 1733, 1787, 1877, 1907, 1913, 1997, 2003, 2039, 2267, 2309, 2339
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Comments

Original title was "Primes p such that Ceiling[p/3] + p is prime." If p = 1 mod 6, then p/3 falls between 2 and 3 mod 6, and the ceiling function bumps it up to 3 mod 6. Therefore ceiling(p/3) + p = 4 mod 6, which is an even number greater than 2 and therefore obviously composite.
Therefore the ceiling function is only necessary when the primality testing function requires an integer argument.
And so, aside from 2, all terms are congruent to 5 mod 6.
Set q = (p + 1)/3 + p, then (p + 1)/(q + 1) = 3/4. If this sequence is proven infinite, that would prove two specific cases of the Schinzel-Sierpiński conjecture regarding rational numbers. - Alonso del Arte, Mar 12 2016

Examples

			2 is in the sequence because (2 + 1)/3 + 2 = 1 + 2 = 3, which is prime.
5 is in the sequence because (5 + 1)/3 + 5 = 2 + 5 = 7, which is prime.
11 is not in the sequence because (11 + 1)/3 + 11 = 15 = 3 * 5.

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714, A158719, A158720, A270384.

Programs

Extensions

Title simplified by Alonso del Arte, Mar 12 2016

A158722 Primes p which are not in A158720 and A158721.

Original entry on oeis.org

3, 7, 11, 19, 29, 37, 41, 43, 47, 61, 71, 79, 83, 89, 97, 101, 107, 109, 127, 131, 137, 139, 151, 157, 163, 173, 191, 199, 223, 227, 229, 239, 241, 251, 257, 263, 271, 277, 281, 283, 293, 311, 313, 317, 331, 349, 353, 367, 373, 379, 383, 389, 397, 401, 409, 419
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714, A158719, A158720, A158721

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[ !PrimeQ[Floor[p/3]+p]&&!PrimeQ[Ceiling[p/3]+p],AppendTo[lst,p]],{n,5!}];lst

A158723 Greater of twin primes in A158720.

Original entry on oeis.org

13, 31, 73, 103, 181, 193, 433, 463, 571, 643, 661, 823, 1021, 1291, 1621, 1723, 2083, 2143, 2341, 2593, 2713, 3001, 3253, 3331, 3361, 3541, 4231, 4243, 4423, 4933, 5233, 5653, 5881, 6553, 6571, 6781, 6871, 6961, 7951, 8293, 9283, 9343, 9433, 9631, 9931
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Comments

If prime number from sequence A158720 is twin prime, it always (?) Greater of twin primes, and none (?) of Lesser of twin primes.

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714, A158719, A158720, A158721, A158722

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[Floor[p/3]+p],If[PrimeQ[p-2],AppendTo[lst,p]]],{n,7!}];lst
Showing 1-6 of 6 results.

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