A023208 -id:A023208 - cp's OEIS frontend

A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

2, 2, 5, 5, 7, 7, 11, 13, 13, 17, 19, 23, 23, 29, 37, 37, 43, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 103, 113, 127, 127, 137, 139, 149, 163, 163, 167, 167, 173, 181, 191, 193, 197, 199, 211, 227, 233, 239, 251, 257, 257, 263, 269, 271, 277, 293

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 start with  7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

Cf. A000040, A023208, A088878, A091180, A094525, A265763.

x = 3; 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, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)

A112391 Primes p such that 23*p + 2 is also prime.

Original entry on oeis.org

3, 7, 19, 37, 43, 67, 103, 157, 193, 277, 337, 367, 373, 379, 397, 439, 457, 463, 547, 607, 829, 859, 877, 919, 1009, 1033, 1039, 1117, 1129, 1213, 1249, 1297, 1429, 1543, 1579, 1627, 1657, 1699, 1723, 1759, 1783, 1789, 1867, 1993, 1999, 2053, 2083, 2089

Offset: 1

Parthasarathy Nambi, Dec 04 2005

nonn

			If p=277 then 23*p + 2 = 6373 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A023208.

[p: p in PrimesUpTo(3000)|  IsPrime(23*p+2)]; // Vincenzo Librandi, Jan 29 2011

Select[Prime[Range[400]],PrimeQ[23#+2]&] (* Harvey P. Dale, Sep 26 2021 *)

forprime(p=2,5000,if(isprime(23*p+2),print1(p,","))) \\ Lambert Herrgesell, Dec 09 2005

More terms from Lambert Herrgesell (zero815(AT)googlemail.com), Dec 09 2005

A370156 Primes p such that (p-2)/3 is prime and 3*p+2 is composite.

Original entry on oeis.org

11, 41, 53, 71, 113, 131, 179, 251, 311, 449, 491, 521, 593, 599, 683, 701, 719, 773, 809, 881, 941, 1049, 1061, 1103, 1151, 1229, 1301, 1319, 1373, 1439, 1499, 1511, 1571, 1709, 1733, 1931, 2273, 2309, 2393, 2579, 2591, 2663, 2843, 2861, 2903, 3041, 3119

Offset: 1

Clark Kimberling, Feb 10 2024

nonn

None of these primes are lucky (A000959). - Davide Rotondo, Feb 12 2025

			(11-2)/3 is a prime and 3*11+2 isn't.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000959, A115058 (supersequence), A023208, A370157.

filter:= proc(p) isprime(p) and isprime((p-2)/3) and not isprime(3*p+2) end proc:
select(filter, [seq(i,i=5..10000,6)]); # Robert Israel, Feb 28 2025

Select[Prime[Range[500]], ! PrimeQ[3 # + 2] && PrimeQ[(# - 2)/3] &]

isok(p) = if (isprime(p), !isprime(3*p+2) && (((p%3) == 2) && isprime((p-2)/3))); \\ Michel Marcus, Feb 17 2024

A023307 Primes that remain prime through 4 iterations of function f(x) = 3x + 2.

Original entry on oeis.org

1129, 10009, 11489, 12539, 13859, 30029, 63079, 77359, 99119, 121039, 124669, 169409, 194749, 205589, 246329, 330329, 349519, 351829, 354839, 361279, 369539, 384589, 395719, 399769, 416989, 429109, 446819, 527599, 532489, 544259, 575119

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+2, 9*p+8, 27*p+26, and 81*p+80 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023208, A023246, A023277, and A024893.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(3*n+2) and IsPrime(9*n+8) and IsPrime(27*n+26) and IsPrime(81*n+80)] // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[50000]],AllTrue[Rest[NestList[3#+2&,#,4]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 13 2015 *)

a(n) == 9 or 69 (mod 70). - John Cerkan, Oct 04 2016

A162338 Primes q such that q = floor(p/3) for some prime p.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 43, 59, 79, 83, 89, 97, 103, 127, 139, 149, 163, 167, 173, 197, 199, 227, 233, 239, 257, 269, 293, 313, 317, 337, 349, 353, 367, 383, 397, 409, 419, 433, 439, 457, 479, 499, 503, 523, 569, 577, 607, 643, 659, 709, 757, 769

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 01 2009

nonn

Primes q such that 3*q+1 or 3*q+2 is prime. Agrees with A023208 except for initial term 2.

Essentially the same as A023208. - R. J. Mathar, Jul 05 2009

			3 is in the sequence since 11 is prime and floor(11/3) = 3; 11 is not in the sequence since 11 = floor(34/3) = floor(35/3) and neither 34 nor 35 is prime.

Cf. A162337. Essentially the same as A023208 (n and 3n+2 are both prime).

[ q: q in PrimesUpTo(800) | IsPrime(3*q+1) or IsPrime(3*q+2) ]; // Klaus Brockhaus, Jul 06 2009

lst={};Do[r=Prime[n];If[PrimeQ[p=Floor[r/3]],AppendTo[lst,p]],{n,6!}];lst
Select[Floor[Prime[Range[350]]/3],PrimeQ] (* Harvey P. Dale, Aug 26 2013 *)
Select[Prime[Range[200]],AnyTrue[3#+{1,2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 07 2019 *)

isA162338(n) = isprime(n) && (isprime(3*n+1) || isprime(3*n+2)) \\ Michael B. Porter, Jan 30 2010

Edited and listed terms verified by Klaus Brockhaus, Jul 06 2009

A163628 Integers such that the two adjacent integers are a prime and three times a prime.

Original entry on oeis.org

8, 10, 14, 16, 20, 22, 32, 38, 40, 52, 58, 68, 70, 88, 110, 112, 128, 130, 140, 158, 178, 182, 200, 212, 238, 250, 268, 292, 308, 310, 338, 380, 382, 410, 418, 448, 488, 490, 500, 502, 520, 542, 572, 578, 592, 598, 632, 682, 700, 718, 752, 770, 772, 788, 808

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2009

nonn

Union[3*A023208 + 1, 3*A088878 - 1]. [Zak Seidov, Aug 07 2009]

			a(1)=8 which lies between 7=A000040(4) and 9 = A001748(2).
a(2)=10 which lies between 9=A001748(2) and 11 = A000040(5).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A163492.

n = 1; A023208 = {}; Do[If[PrimeQ[(Prime[k] - 2 n)/(2 n + 1)], AppendTo[A023208, (Prime[k] - 2 n)/(2 n + 1)]], {k, 1, 1000}]; A023208;
A088878 = {}; Do[p = Prime[n]; If[PrimeQ[3*p - 2], AppendTo[A088878, p]], {n, 5!}]; A088878; Union[3*A023208 + 1, 3*A088878 - 1] (* G. C. Greubel, Jul 30 2017 *)

Many terms like 44, 46, 62 etc. removed by R. J. Mathar, Aug 06 2009

A370157 Primes p such that both 3p+2 and (p-2)/3 are composite or 0.

Original entry on oeis.org

2, 31, 47, 61, 67, 73, 101, 107, 109, 137, 151, 157, 181, 191, 193, 211, 223, 229, 241, 263, 271, 277, 281, 283, 307, 331, 347, 359, 373, 379, 389, 401, 421, 431, 443, 461, 463, 467, 487, 509, 541, 547, 557, 563, 571, 587, 601, 613, 617, 619, 631, 641, 647

Offset: 1

Clark Kimberling, Feb 10 2024

nonn

Cf. A000040, A115058 (supersequence), A023208, A370156.

Select[Prime[Range[200]], ! PrimeQ[3 # + 2] && ! PrimeQ[(# - 2)/3] &]

isok(p) = if (isprime(p), !isprime(3*p+2) && !(((p%3) == 2) && isprime((p-2)/3))); \\ Michel Marcus, Feb 17 2024

A120637 Primes such that their triple is 2 away from a prime number.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 113, 127, 137, 139, 149, 163, 167, 173, 181, 191, 193, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 271, 277, 293, 307, 313, 317, 331, 337, 347, 349, 353, 367, 373, 383

Offset: 1

Cino Hilliard, Aug 17 2006

This sequence is a variation of the sequence in the reference. However, this sequence should have an infinite number of terms.

			19 is a prime and 19*3 = 57 which is two away from 59 which is prime.
31 is not in the table because 31*3 = 93 which is 2 away from 91 and 95, both not prime.

R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective, Springer Verlag 2002, p. 49, exercise 1.18.

Cf. A125272.

Select[Prime[Range[200]],PrimeQ[3#+2]||PrimeQ[3#-2]&] (* Harvey P. Dale, Aug 10 2011 *)

primepm2(n,k) { local(x,p1,p2,f1,f2,r); if(k%2,r=2,r=1); for(x=1,n, p1=prime(x); p2=prime(x+1); if(isprime(p1*k+r)||isprime(p1*k-r), print1(p1",") ) ) }

Union of A023208 and A088878.

A120638 Primes such that their triple is not 2 away from a prime number.

Original entry on oeis.org

2, 31, 41, 73, 101, 107, 109, 131, 151, 157, 179, 223, 229, 241, 281, 283, 311, 359, 379, 389, 421, 449, 463, 509, 521, 547, 563, 571, 599, 613, 617, 619, 631, 641, 647, 653, 661, 683, 691, 701, 719, 733, 739, 743, 773, 787, 809, 811, 821, 827, 829, 839, 857

Offset: 1

Cino Hilliard, Aug 17 2006

This sequence is a variation of the sequence in the reference. However, this sequence should have an infinite number of terms. k=2 in the PARI code.

			31*3 = 93 which is two away from 91 and 95 both not prime.

R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective, Springer Verlag 2002, p. 49, exercise 1.18.

Cf. A023208, A088878, A125272.

Select[Prime@Range@200,!PrimeQ[3#-2]&&!PrimeQ[3#+2]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)
Select[Prime[Range[200]],NoneTrue[3#+{2,-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 01 2019 *)

primepm3(n,k) = =number of iterations,k = factor { local(x,p1,p2,f1,f2,r); if(k%2,r=2,r=1); for(x=1,n, p1=prime(x); p2=prime(x+1); if(!isprime(p1*k+r)&!isprime(p1*k-r), print1(p1",") ) ) }

A348307 Primes p such that (p-1)/2, (p-2)/3, 2p+1, 3p+2 are all prime numbers.

Original entry on oeis.org

23, 21383, 26459, 28643, 111263, 137339, 217643, 333563, 342599, 423323, 486023, 540539, 548519, 567719, 658943, 671039, 755663, 829463, 865499, 890063, 903803, 976883, 1108259, 1168523, 1199183, 1308383, 1316699, 1318379, 1342403, 1349423, 1390199, 1501583, 1503059, 1558079, 1563119

Offset: 1

Marc Morgenegg, Oct 11 2021

nonn

For (p-1)/2, those are the safe primes A005385.

			23 is a term because: (23-1)/2 = 11, (23-2)/3 = 7, 2*23+1 = 47, 3*23+2 = 71, {23, 11, 7, 47, 71} are all prime numbers.

Cf. A005385 (safe primes).

Intersection of A005385 and A094524 and A005384 and A023208.

Select[Range[1, 1.5*10^6, 2], AllTrue[{#, (# - 1)/2, (# - 2)/3, 2*# + 1, 3*# + 2}, PrimeQ] &] (* Amiram Eldar, Oct 11 2021 *)

isok(p) = iferr(isprime(p) && isprime((p-1)/2) && isprime((p-2)/3) && isprime(2*p+1) && isprime(3*p+2), E, 0); \\ Michel Marcus, Oct 11 2021

More terms from Michel Marcus, Oct 11 2021

cp's OEIS Frontend

A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

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Mathematica

A112391 Primes p such that 23*p + 2 is also prime.

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Author

Keywords

Examples

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Programs

Magma

Mathematica

PARI

Extensions

A370156 Primes p such that (p-2)/3 is prime and 3*p+2 is composite.

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Maple

Mathematica

PARI

A023307 Primes that remain prime through 4 iterations of function f(x) = 3x + 2.

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Magma

Mathematica

Formula

A162338 Primes q such that q = floor(p/3) for some prime p.

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Magma

Mathematica

PARI

Extensions

A163628 Integers such that the two adjacent integers are a prime and three times a prime.

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Comments

Examples

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Crossrefs

Programs

Mathematica

Extensions

A370157 Primes p such that both 3p+2 and (p-2)/3 are composite or 0.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A120637 Primes such that their triple is 2 away from a prime number.

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Author

Keywords

A348307 Primes p such that (p-1)/2, (p-2)/3, 2p+1, 3p+2 are all prime numbers.