A153183 -id:A153183 - cp's OEIS frontend

A088878 Prime numbers p such that 3p - 2 is a prime.

3, 5, 7, 11, 13, 23, 37, 43, 47, 53, 61, 67, 71, 103, 113, 127, 137, 163, 167, 181, 191, 193, 211, 251, 257, 263, 271, 277, 293, 307, 313, 331, 337, 347, 373, 401, 431, 433, 443, 461, 467, 487, 491, 523, 541, 557, 587, 593, 601, 673, 677, 727, 751, 757, 761

Offset: 1

Giovanni Teofilatto, Nov 27 2003

Indices of semiprime octagonal numbers. - Jonathan Vos Post, Feb 16 2006
Daughter primes of order 1. - Artur Jasinski, Dec 12 2007
A010051(3*a(n)-2) = 1. - Reinhard Zumkeller, Jul 02 2015

			For p = 3, 3p - 2 = 7;
for p = 523, 3p - 2 = 1567.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Octagonal Number.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A000040, A000567, A001222, A001358, A091179, A091180, A091181, A136019, A136020, A153183, A153184.
Cf. A010051, A063908, A172287.
Cf. A259730.

a088878 n = a088878_list !! (n-1)
a088878_list = filter ((== 1) . a010051' . subtract 2 . (* 3)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[ p: p in PrimesUpTo(770) | IsPrime(3*p-2) ]; // Klaus Brockhaus, Dec 21 2008

lst={};Do[p=Prime[n];If[PrimeQ[3*p-2],AppendTo[lst,p]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 22 2008 *)
n = 1; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, (Prime[k] + 2n)/(2n + 1)]], {k, 1, 500}]; a (* Artur Jasinski, Dec 12 2007 *)
Select[Prime[Range[150]],PrimeQ[3#-2]&] (* Harvey P. Dale, Feb 27 2024 *)

list(lim)=select(p->isprime(3*p-2),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

Corrected and extended by Ray Chandler, Dec 27 2003

Entry revised by N. J. A. Sloane, Nov 28 2006, Jul 08 2010

A034936 Numbers k such that 3*k + 4 is prime.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 19, 21, 23, 25, 31, 33, 35, 41, 45, 49, 51, 53, 59, 63, 65, 69, 73, 75, 79, 89, 91, 93, 101, 103, 109, 111, 115, 121, 123, 125, 131, 135, 139, 143, 145, 151, 153, 161, 165, 173, 179, 181, 189, 191, 199, 201, 203, 205, 209, 213, 219, 223, 229

Offset: 1

Jud McCranie

Related to hyperperfect numbers of a certain form.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.

Cf. A038536 and A002476.
A002476 gives primes, A091178 gives prime index.
a(n) = A024892(n) - 1 = 2*A024899(n) - 1.
a(n) = A153183(n) - 2 = A107303(n) - 3.

[n: n in [1..1000] |IsPrime(3*n+4)] // Vincenzo Librandi, Nov 17 2010

lst={};Do[p=n+(n+1)+(n+3);If[PrimeQ[p],AppendTo[lst,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 22 2009 *)
Select[Range[300],PrimeQ[3#+4]&] (* Harvey P. Dale, Aug 25 2016 *)

is(n)=isprime(3*n+4) \\ Charles R Greathouse IV, Jul 02 2013

A024892 Numbers k such that 3*k+1 is prime.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 20, 22, 24, 26, 32, 34, 36, 42, 46, 50, 52, 54, 60, 64, 66, 70, 74, 76, 80, 90, 92, 94, 102, 104, 110, 112, 116, 122, 124, 126, 132, 136, 140, 144, 146, 152, 154, 162, 166, 174, 180, 182, 190, 192, 200, 202, 204, 206, 210, 214, 220, 224, 230, 236, 242, 244, 246

Offset: 1

Clark Kimberling

Every prime (with the exception of 3) can be expressed as 3*k+1 or 3*k-1. - César Aguilera, Apr 13 2013
The associated prime A002476(n) has a unique representation as x^2 + x*y - 2*y^2 = (x + 2*y)*(x-y) with positive integers, namely (x(n), y(n)) = (a(n) + 1, a(n)). See the N. J. A. Sloane, May 31 2014, comment on A002476. - Wolfdieter Lang, Feb 09 2016
For all elements of this sequence there are no (x,y) positive integers such that a(n) = 3*x*y + x + y or a(n) = 3*x*y - x - y. - Pedro Caceres, Jan 28 2021

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

Cf. A002476 (associated primes), A091178 (gives prime index).

Cf. A153183, A107303.

[n: n in [1..1000] | IsPrime(3*n+1)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[250], PrimeQ[3# + 1] &] (* Vincenzo Librandi, Sep 25 2012 *)

is(n)=isprime(3*n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A002476(n) - 1)/3. See the name.
a(n) = 2*A024899(n) = A034936(n) + 1.
a(n) = A153183(n) - 1 = A107303(n) - 2.

A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

Original entry on oeis.org

3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127, 133, 147, 153, 155, 163, 167, 205, 215, 225, 247, 253, 257, 275, 285, 287, 293, 295, 303, 313, 323, 337, 363, 365, 405, 427, 433, 435, 475, 477, 483, 495, 497, 517

Offset: 1

Kyle D. Balliet, Mar 07 2009

nonn

Barycenter of cousin primes (A029708; see also A029710, A023200, A046132), divided by 3. When p>3 and p+4 both are prime, then p = 1 (mod 6) and p+2 = 3 (mod 6). - M. F. Hasler, Jan 14 2013

			15*3 +/- 2 = 43,47 (both prime).

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

Intersection of A024893 and A153183.

[n: n in [1..1000]|IsPrime(3*n-2)and IsPrime(3*n+2)] // Vincenzo Librandi, Dec 13 2010

select(t -> isprime(3*t+2) and isprime(3*t-2), [seq(t,t=3..1000,2)]); # Robert Israel, May 28 2017

Select[Range[600],AllTrue[3#+{2,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2019 *)

Intersection of A024893 and A153183.
a(n) = A029708(n)/3. - Zak Seidov, Aug 07 2009
a(n) = A056956(n)*2+1 = (A029710(n)+2)/3 = (A023200(n+1)+2)/3. - M. F. Hasler, Jan 14 2013

A228121 Numbers n such that 3n - 4 is prime.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 15, 17, 19, 21, 25, 29, 31, 35, 37, 39, 45, 47, 51, 57, 59, 61, 65, 67, 77, 79, 81, 85, 87, 89, 91, 95, 99, 105, 107, 117, 119, 121, 129, 131, 135, 141, 145, 149, 151, 155, 157, 161, 165, 169, 171, 175, 187, 189, 191, 197, 199, 201, 207, 215, 217, 219, 221, 227, 229

Offset: 1

Irina Gerasimova, Aug 11 2013

Primes in a(n): 2, 3, 5, 7, 11, 17, 19, 29, 31, 37, 47, 59, 61, 67, 79, 89, 107, 131, 149, 151, 157, 191, 197, 199, 227, 229, 241, 271, 277, 281,...

			For n = 15, 3*15 - 4 = 41 is prime.

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

Cf. A008585, A153183, A228358.

Select[Range[300],PrimeQ[3#-4]&] (* Harvey P. Dale, Mar 30 2020 *)

is(n)=isprime(3*n-4) \\ Charles R Greathouse IV, Mar 17 2014

a(n) = A024893(n) + 2. - Michael B. Porter, Aug 11 2013

A107303 Numbers k such that (3*k - 5) is prime.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 44, 48, 52, 54, 56, 62, 66, 68, 72, 76, 78, 82, 92, 94, 96, 104, 106, 112, 114, 118, 124, 126, 128, 134, 138, 142, 146, 148, 154, 156, 164, 168, 176, 182, 184, 192, 194, 202, 204, 206, 208, 212, 216, 222, 226, 232

Offset: 1

Parthasarathy Nambi, May 20 2005

3 and 5 are twin primes.

			If k=4, then 3*k - 5 = 7 (prime).
If k=28, then 3*k - 5 = 79 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A088879.
Cf. A153183, A024892, A034936.
Equals A153183(n) + 1; also A024892(n) + 2; also A034936(n) + 3;

[n: n in [2..300] | IsPrime(3*n-5)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[2,250],PrimeQ[3#-5]&] (* Harvey P. Dale, Mar 31 2024 *)

is(n)=isprime((3*n-5)) \\ Charles R Greathouse IV, Jun 12 2017

A153184 Numbers n such that 3*n-2 is not prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 34, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 59, 60, 62, 63, 64, 66, 68, 69, 70, 72, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 94, 96, 97, 98, 99, 100

Offset: 1

Vincenzo Librandi, Dec 20 2008

One more than the associated value in A153309. - R. J. Mathar, Jan 05 2011

			Distribution of the odd terms > a(1) in the following triangular array:
*;
*,9;
*,*,17;
*,*,*,*;
*,19,*,*,41;
*,*,31,*,*,57;
*,*,*,*,*,*,*;
*,29,*,*,63,*,*,97;
*,*,45,*,*,83,*,*,121;
*,*, *,*,*,*, *,*, *, *;
*,39,*,*,85,*,*,131,*,*,177;
*,*,59,*,*,109,*,*,159,*,*,209; etc.
where * marks the non-integer values of (4*h*k + 2*k + 2*h + 3)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

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

Cf. A153183, A088878.

[n: n in [0..120] | not IsPrime(3*n - 2)]; // Vincenzo Librandi, Jan 12 2013

lst={};Do[If[ !PrimeQ[3*n-2],AppendTo[lst,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 22 2008 *)
Select[Range[0, 100], !PrimeQ[3 # - 2] &] (* Vincenzo Librandi, Jan 12 2013 *)

is(n)=!isprime(3*n-2) \\ Charles R Greathouse IV, Oct 26 2015

a(n) ~ n. - Charles R Greathouse IV, Oct 26 2015

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

A129926 Semiprimes s such that 3*s - 2 is a prime.

Original entry on oeis.org

15, 21, 25, 33, 35, 51, 55, 65, 77, 91, 93, 95, 111, 123, 133, 141, 145, 155, 183, 201, 203, 205, 215, 221, 237, 247, 253, 287, 295, 303, 323, 341, 355, 365, 377, 391, 411, 413, 417, 427, 485, 497, 511, 515, 517, 527, 533, 537, 543, 553, 565, 581, 583, 597

Offset: 1

Giovanni Teofilatto, Jun 06 2007

nonn

Indices of 3-almost prime octagonal numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A153183 and A001358 (semiprimes).

isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true ; else false ; end ; end: isA129926 := proc(n) if isA001358(n) then isprime(3*n-2) ; else false ; fi ; end: for n from 1 to 1000 do if isA129926(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jun 07 2007

Select[Range[600],PrimeOmega[#]==2&&PrimeQ[3#-2]&] (* James C. McMahon, Feb 02 2025 *)

list(lim)=my(v=List(),n); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), n=p*q; if(isprime(3*n-2), listput(v,n)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

More terms from R. J. Mathar, Jun 07 2007

A268475 Numbers n such that n^3 +/- 2 and 3*n +/- 2 are all prime.

Original entry on oeis.org

435, 555, 2415, 31635, 38025, 44835, 80625, 88335, 97455, 98505, 99435, 124335, 142065, 145095, 165375, 176055, 204765, 246435, 279225, 293475, 310095, 315555, 332085, 344745, 348735, 376935, 392415, 443595, 462105, 467385, 482355, 581415, 609555, 626775, 636015

Offset: 1

K. D. Bajpai, Feb 05 2016

nonn

All the terms in this sequence are congruent to 0 (mod 5).
Each term in this sequence yields two sets of cousin prime pairs i.e., for n = 435 -> {82312877, 82312873} and {1307, 1303}.
All terms are congruent to 15 mod 30. - Robert Israel, Feb 05 2016

			435 is in the sequence because 435^3 + - 2 =  82312877, 82312873; 3*435 + - 2 = 1307, 1303 are all prime.
555 is in the sequence because 555^3 + - 2 =  170953877, 170953873; 3*555 + - 2 = 1667, 1663 are all prime.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A024893, A090121, A108701, A153183, A157834.

[n : n in [1..1e5] | IsPrime(n^3 + 2) and IsPrime(n^3 - 2) and IsPrime(3*n + 2) and IsPrime(3*n - 2)];

select(n -> andmap(isprime, [n^3 + 2, n^3 - 2, 3*n + 2, 3*n - 2]), [seq(p, p=1.. 10^6)]);

Select[Range[1000000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[3 # + 2] && PrimeQ[3 # - 2] &]

for(n = 1,1e5, if( isprime(n^3 + 2) && isprime(n^3 - 2) && isprime(3*n + 2) && isprime(3*n - 2), print1(n ", ")))

A294064 Numbers k such that 2k - 3, 2k + 3, 3k - 2, 3k + 2 are primes.

Original entry on oeis.org

5, 7, 13, 35, 43, 55, 77, 127, 133, 155, 167, 253, 287, 295, 365, 475, 497, 533, 595, 713, 1007, 1177, 1483, 1805, 2323, 2575, 2723, 2927, 3107, 3415, 3487, 3823, 4145, 4213, 4367, 4565, 4717, 4927, 4963, 5125, 5215, 5363, 5417, 5587, 5627, 5795, 6133, 6587, 6797

Offset: 1

Dimitris Valianatos, Oct 22 2017

nonn

The common numbers of A098090, A067076, A153183, A024893.
Conjecture: The Sum_{n>=1} 1/a(n) = 0.57... converges.
Note that the sum of the 4 primes that are obtained is 10 times the original term: (2*k - 3) + (2*k + 3) + (3*k - 2) + (3*k + 2) = 10*k.
From Robert G. Wilson v, Nov 19 2017: (Start)
Number of terms less than 10^m: 2, 7, 20, 55, 189, 919, 4863, 28218, 174469, ..., ;
Number of prime terms less than 10^m: 2, 4, 6, 12, 39, 140, 558, 2755, 14804, ..., .
All terms are == {5, 7, 13, 17, 23, 25} (mod 30).
(End)

			5 is in the sequence because 2*5-3 = 7, 2*5+3 = 13, 3*5-2 = 13, 3*5+2 = 17 and the tetrad [7, 13, 13, 17] are all prime numbers.
7 is in the sequence because 2*7-3 = 11, 2*7+3 = 17, 3*7-2 = 19, 3*7+2 = 23 and the tetrad [11, 17, 19, 23] are all prime numbers.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A098090, A067076, A153183, A024893.

Select[Range[10^4], Function[k, AllTrue[Flatten@ Map[#1 k + {-1, 1} #2 & @@ # &, {#, Reverse@ #}] &@ {2, 3}, PrimeQ]]] (* Michael De Vlieger, Oct 22 2017 *)

{
for(n=1,10000,
    if(isprime(2*n-3)&&isprime(2*n+3)&&isprime(3*n-2)&&isprime(3*n+2),
       print1(n", ")
      )
   )
}

cp's OEIS Frontend

A088878 Prime numbers p such that 3p - 2 is a prime.

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A034936 Numbers k such that 3*k + 4 is prime.

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A024892 Numbers k such that 3*k+1 is prime.

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A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

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A228121 Numbers n such that 3n - 4 is prime.

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A107303 Numbers k such that (3*k - 5) is prime.

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A153184 Numbers n such that 3*n-2 is not prime.

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A294064 Numbers k such that 2k - 3, 2k + 3, 3k - 2, 3k + 2 are primes.