A158719 -id:A158719 - cp's OEIS frontend

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

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

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

nonn

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

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

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

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

nonn

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

			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.

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

Select[Prime[Range[350]], PrimeQ[(# + 1)/3 + #] &] (* Harvey P. Dale, Feb 24 2013, simplified by Alonso del Arte, Mar 12 2016 *)

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

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

nonn

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

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

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

nonn

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

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

lst={};Do[p=Prime[n];If[PrimeQ[Floor[p/3]+p],If[PrimeQ[p-2],AppendTo[lst,p]]],{n,7!}];lst

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A158722 Primes p which are not in A158720 and A158721.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A158723 Greater of twin primes in A158720.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica