A158708 -id:A158708 - 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

A164621 Primes p such that pfloor(p/2)-2 and pfloor(p/2)+2 are also prime numbers.

Original entry on oeis.org

7, 31, 79, 211, 271, 751, 787, 1231, 1447, 1459, 2347, 2551, 3727, 5119, 6427, 6691, 8467, 8707, 9007, 10099, 10531, 10567, 10831, 11959, 18691, 21487, 22039, 22567, 23059, 23167, 23371, 24379, 24499, 25171, 26371, 27967, 28579, 28591, 29287

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

nonn

			7*3-2=13, 7*3+2=17,..

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A068229, A158708, A164620

lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]-2]&&PrimeQ[p*Floor[p/2]+2],AppendTo[lst,p]],{n,2*7!}];lst
Select[Prime[Range[3200]],AllTrue[# Floor[#/2]+{2,-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 04 2020 *)

A265761 Numerators of primes-only best approximates (POBAs) to 3/2; see Comments.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 43, 47, 61, 71, 79, 89, 101, 107, 109, 151, 163, 191, 197, 223, 227, 251, 269, 271, 317, 349, 359, 421, 439, 461, 467, 521, 523, 569, 601, 613, 631, 647, 659, 673, 691, 701, 719, 811, 821, 853, 857, 881, 911, 919, 929, 947, 971, 991

Offset: 1

Clark Kimberling, Dec 18 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

			The POBAs for 3/2 start with 2/2, 5/3, 7/5, 11/7, 17/11, 19/13, 29/19, 43/29, 47/31. For example, if p and q are primes and q > 13, then 19/13 is closer to 3/2 than p/q is.

Cf. A000040, A104163, A158708, A158790, A162336, A222565.

x = 3/2; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265761/A222565 *)
Numerator[tL]   (* A104163 *)
Denominator[tL] (* A158708 *)
Numerator[tU]   (* A162336 *)
Denominator[tU] (* A158709 *)
Numerator[y]    (* A265761 *)
Denominator[y]  (* A222565 *)

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

A164622 Primes p such that pfloor(p/2) - 4 and pfloor(p/2) + 4 are prime numbers.

Original entry on oeis.org

151, 463, 571, 631, 643, 991, 1063, 1171, 1831, 2083, 2311, 4951, 5023, 6211, 6703, 6763, 7723, 7951, 9043, 11383, 12163, 12391, 13183, 14851, 15031, 17431, 19231, 19543, 20143, 22051, 23143, 25951, 26371, 27283, 28351, 29131, 30643, 32803

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

nonn

151*75-4=11321 (prime), 151*75+4=11329 (prime), ..

Subsequence of A068229. Cf. A008846, A020882, A068229, A086519, A158708, A164620, A164621

lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]-4]&&PrimeQ[p*Floor[p/2]+4],AppendTo[lst,p]],{n,8!}];lst

Edited by Charles R Greathouse IV, Nov 02 2009

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.

Original entry on oeis.org

13, 157, 673, 1069, 1117, 1153, 1213, 1597, 2029, 2089, 2437, 2713, 2833, 3613, 4057, 4909, 5653, 6337, 6529, 7549, 7993, 8053, 9613, 10789, 11497, 11689, 12073, 12373, 13309, 13669, 13789, 14173, 15289, 15937, 16249, 18097, 18637, 19249, 19993

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Primes A000040(k) such that A008837(k)+-5 are also prime numbers.

			13 is in the sequence because 13*6-5=73 and 13*6+5=83 are both prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008846, A020882, A068228, A068229, A082539, A086519, A107159, A158708, A139494, A164620, A164621, A164622, A023203.

Select[Prime[Range[2300]], PrimeQ[# (# - 1)/2 - 5] && PrimeQ[# (# - 1)/2 + 5] &]

forprime(p=2,10^6,my(b=binomial(p,2));if(isprime(b-5)&isprime(b+5),print1(p,", "))); /* Joerg Arndt, Apr 10 2013 */

Edited by R. J. Mathar, Aug 20 2009

Mathematica code adapted to the definition by Bruno Berselli, Apr 10 2013

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

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

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

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A164621 Primes p such that p*floor(p/2)-2 and p*floor(p/2)+2 are also prime numbers.

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A265761 Numerators of primes-only best approximates (POBAs) to 3/2; see Comments.

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A158722 Primes p which are not in A158720 and A158721.

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A164622 Primes p such that p*floor(p/2) - 4 and p*floor(p/2) + 4 are prime numbers.

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A164623 Primes p such that p*(p-1)/2-5 and p*(p-1)/2+5 are also prime numbers.

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A158723 Greater of twin primes in A158720.

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A164621 Primes p such that pfloor(p/2)-2 and pfloor(p/2)+2 are also prime numbers.

A164622 Primes p such that pfloor(p/2) - 4 and pfloor(p/2) + 4 are prime numbers.

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.