A259730 -id:A259730 - 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

A063908 Numbers k such that k and 2*k-3 are primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 53, 67, 71, 83, 97, 101, 107, 113, 127, 137, 157, 167, 181, 191, 193, 211, 223, 233, 241, 251, 263, 283, 311, 317, 331, 347, 373, 421, 431, 433, 443, 457, 461, 487, 521, 547, 563, 577, 587, 613, 617, 631, 641, 643, 647

Offset: 1

N. J. A. Sloane, Aug 31 2001

nonn

If p is in this sequence then the products of positive powers of 3, p and 2p-3 are entries in A086486. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Median prime of AP3's starting at 3, i.e., triples of primes (3,p,q) in arithmetic progression. - M. F. Hasler, Sep 24 2009
a(n) = sum of the coprimes(p) mod (p+1). - J. M. Bergot, Nov 13 2014
A010051(2*a(n)-3) = 1. - Reinhard Zumkeller, Jul 02 2015
A098090 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

			From _K. D. Bajpai_, Nov 29 2019: (Start)
a(5) = 13 is prime and 2*13 - 3 = 23 is also prime.
a(6) = 17 is prime and 2*17 - 3 = 31 is also prime.
(End)

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 34.- Prime Triplets in arithmetic progression, The Prime Puzzles & Problems Connection. [From _M. F. Hasler_, Sep 24 2009]

Cf. A005382.
Cf. A000040, A010051, A088878, A172287.
Cf. A259730.

a063908 n = a063908_list !! (n-1)
a063908_list = filter
   ((== 1) . a010051' . (subtract 3) . (* 2)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[n : n in [0..700] | IsPrime(n) and IsPrime(2*n-3)]; // Vincenzo Librandi, Nov 14 2014

select(k -> andmap(isprime, [k, 2*k-3]), [seq(k, k=1.. 10^4)]); # K. D. Bajpai, Nov 29 2019

Select[Prime[Range[6! ]],PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

{ n=0; p=1; for (m=1, 10^9, p=nextprime(p+1); if (isprime(2*p - 3), write("b063908.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 02 2009

forprime( p=1,default(primelimit), isprime(2*p-3) && print1(p",")) \\ M. F. Hasler, Sep 24 2009

a(n) = A241817(n)/2. - Wesley Ivan Hurt, Apr 08 2018

A172287 Primes p such that exactly one of 2p-3 and 3p-2 is prime.

Original entry on oeis.org

17, 31, 41, 47, 61, 83, 97, 101, 103, 107, 157, 163, 223, 233, 241, 257, 271, 277, 283, 293, 307, 311, 313, 317, 337, 401, 421, 457, 467, 491, 521, 523, 541, 547, 557, 563, 577, 593, 601, 613, 617, 631, 641, 643, 647, 653, 661, 673, 677, 701, 743, 761, 773

Offset: 1

Juri-Stepan Gerasimov, Jan 30 2010

A010051(2*a(n)+3) + A010051(3*a(n)+2) = 1; each term is either a term of A063908 or of A088878. - Reinhard Zumkeller, Jul 02 2015

No terms end in 9. Dickson's conjecture implies that there are infinitely many terms. - Robert Israel, Jul 02 2015

			a(1)=17 because 2*17-3=31 is prime and 3*17-2=49 is nonprime.
19 is not a term because neither 2*19-3=35 nor 3*19-2=55 is prime;
23 is not a term because both 2*23-3=43 and 3*23-2=67 are prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A010051, A063908, A088878, A259730.

a172287 n = a172287_list !! (n-1)
a172287_list = filter
   (\p -> a010051' (2 * p - 3) + a010051' (3 * p - 2) == 1) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

A172287:=n->`if`(isprime(n) and (isprime(2*n-3) xor isprime(3*n-2)), n, NULL): seq(A172287(n), n=1..1000); # Wesley Ivan Hurt, Jun 23 2015

Select[Prime@ Range@ 150, Xor[PrimeQ[2 # - 3], PrimeQ[3 # - 2]] &] (* Michael De Vlieger, Jul 01 2015 *)

Extended by Charles R Greathouse IV, Mar 25 2010

A259758 Squarefree semiprimes of the form (2p - 3) (3*p - 2), p prime.

Original entry on oeis.org

21, 91, 209, 589, 851, 2881, 7739, 10541, 16171, 26069, 29329, 75151, 95129, 110839, 165169, 194219, 216409, 220991, 264389, 374749, 411601, 653069, 717949, 829931, 1108969, 1119311, 1171741, 1269139, 1416689, 2059789, 3161729, 3374249, 3428459, 4924109

Offset: 1

Reinhard Zumkeller, Jul 05 2015

nonn

a(n) = (2*A259730(n) - 3) * (3*A259730(n) - 2);

3431 = A033569(24) = (2*25-3)*(3*25-2) = 47*73 = A006881(946) is the smallest term in the intersection of A006881 and A033569 not belonging to this sequence.

			.    n | p = A259730(n) | 2*p - 3 | 3*p - 2 |   a(n)
.  ----+----------------+---------+---------+--------
.    1 |              3 |       3 |       7 |     21
.    2 |              5 |       7 |      13 |     91
.    3 |              7 |      11 |      19 |    209
.    4 |             11 |      19 |      31 |    589
.    5 |             13 |      23 |      37 |    851
.    6 |             23 |      43 |      67 |   2881
.    7 |             37 |      71 |     109 |   7739
.    8 |             43 |      83 |     127 |  10541
.    9 |             53 |     103 |     157 |  16171
.   10 |             67 |     131 |     199 |  26069
.   11 |             71 |     139 |     211 |  29329
.   12 |            113 |     223 |     337 |  75151  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A259730, subsequence of A006881, subsequence of A033569.

a259758 n = (2 * p - 3) * (3 * p - 2)  where p = a259730 n

Select[Table[(2p-3)(3p-2),{p,Prime[Range[200]]}],PrimeOmega[#]==2&&SquareFreeQ[ #]&] (* Harvey P. Dale, Jul 20 2022 *)

a(n) = 6*A259730(n)^2 - 13*A259730(n) + 6.

A290164 Primes p such that both 4p - 3 and 3p - 4 are prime.

Original entry on oeis.org

2, 5, 11, 19, 29, 59, 61, 79, 89, 131, 149, 151, 191, 389, 431, 479, 499, 521, 541, 571, 631, 659, 701, 739, 919, 941, 971, 1069, 1181, 1279, 1289, 1361, 1381, 1451, 1471, 1489, 1669, 1949, 2069, 2089, 2131, 2549, 2609, 2749, 2791, 3011, 3109, 3181, 3251, 3361

Offset: 1

David James Sycamore, Jul 22 2017

nonn

For n >= 3, all terms end in 1 or 9. - Robert Israel, Jul 24 2017

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

Cf. A259730, A290163 (a subsequence).

select(p -> isprime(p) and isprime(4*p-3) and isprime(3*p-4), [2,seq(i,i=3..10000,2)]); # Robert Israel, Jul 24 2017

Select[Prime@ Range@ 500, Times @@ Boole@ Map[PrimeQ, {4 # - 3, 3 # - 4}] > 0 &] (* Michael De Vlieger, Jul 23 2017 *)

forprime(p=2, 10000, if (isprime(3*p-4) && isprime(4*p-3), print1(p, ", "))) \\ Michel Marcus, Jul 23 2017

More terms from Michel Marcus, Jul 23 2017

A290163 Primes p such that A288814(4p) - A288814(3p) = 7.

Original entry on oeis.org

2, 19, 29, 59, 79, 89, 131, 149, 151, 389, 479, 499, 521, 571, 631, 659, 701, 739, 919, 941, 971, 1069, 1279, 1289, 1361, 1381, 1451, 1471, 1489, 1669, 1949, 2069, 2089, 2131, 2549, 2609, 2749, 2791, 3011, 3109, 3181, 3251, 3361, 3389, 3539, 3581, 3659, 4049, 4091, 4139

Offset: 1

David James Sycamore, Jul 22 2017

nonn

Proper subset of A290164.

Terms of A290164 not in this sequence include 5, 11, 61, 191, 431, 541, 1181, 3571, ... corresponding to primes p such that A(4*p) - A(3*p) = A(3*p) - 1, where A=A288814. Examples: A(4*5) - A(3*5) = 51 - 26 = 25; A(4*541) - A(3*541) = 6483 - 3242 = 3241.

			A288814(4*2) - A288814(3*2) = 15 - 8 = 7, therefore prime 2 is in the sequence;
A288814(4*19) - A288814(3*19) = 219 - 212 = 7, therefore prime 19 is a term.

Cf. A259730, A288814, A290164.

With[{s = Array[Boole[CompositeQ@ #] Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#]] &, 10^5]}, Select[Prime@ Range[600], Function[p, FirstPosition[s, ?(# == 4 p &)][[1]] - FirstPosition[s, ?(# == 3 p &)][[1]] == 7]]] (* Michael De Vlieger, Jul 23 2017 *)

More terms from Altug Alkan, Jul 23 2017

Edited by Robert Israel, Jul 24 2017

A289556 Primes p such that both 5p - 4 and 4p - 5 are prime.

Original entry on oeis.org

3, 7, 13, 43, 67, 109, 127, 151, 163, 211, 277, 307, 373, 457, 463, 601, 613, 673, 727, 853, 919, 967, 1021, 1117, 1171, 1231, 1399, 1471, 1483, 1747, 1789, 1933, 2029, 2251, 2311, 2389, 2503, 2521, 2557, 2659, 2851, 2857, 3019, 3067, 3121, 3229, 3583, 3613, 3637, 3691, 3697

Offset: 1

David James Sycamore, Aug 02 2017

nonn

The terms of this sequence belong to two disjoint subsequences, namely those for which |A(5*p) - A(4*p)| = 9; (3,7,13,43,67,127,163,211,277,307,457,...), and those for which 5*A(4*p) - 3*A(5*p) = 3, (109,151,373,673,919,...), where A = A288814.

Note: A288814(n) = A056240(n) for all composite n.

			P=7: 5*7 - 4 = 31, 4*7 - 5 = 23, both prime so 7 is in this sequence, and belongs to the subsequence of terms satisfying A(4*p) - A(3*p) = 9.
P=109: 5*109 - 4 = 541, 4*109 - 5 = 431, both prime so 109 is in this sequence, and belongs to the subsequence of terms satisfying 5*A(4*p) - 3*A(5*p) = 3.

Cf. A259730, A288814, A290163, A290164, A056240.

Intersection of A136051 and A156300. - Michel Marcus, Aug 04 2017

Select[Prime@ Range@ 516, Times @@ Boole@ Map[PrimeQ, {5 # - 4, 4 # - 5}] > 0 &] (* Michael De Vlieger, Aug 02 2017 *)

More terms from Altug Alkan, Aug 02 2017

cp's OEIS Frontend

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

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A063908 Numbers k such that k and 2*k-3 are primes.

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A172287 Primes p such that exactly one of 2p-3 and 3p-2 is prime.

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A259758 Squarefree semiprimes of the form (2*p - 3) * (3*p - 2), p prime.

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A290164 Primes p such that both 4*p - 3 and 3*p - 4 are prime.

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Maple

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PARI

Extensions

A290163 Primes p such that A288814(4*p) - A288814(3*p) = 7.

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Mathematica

Extensions

A289556 Primes p such that both 5*p - 4 and 4*p - 5 are prime.

A259758 Squarefree semiprimes of the form (2p - 3) (3*p - 2), p prime.

A290164 Primes p such that both 4p - 3 and 3p - 4 are prime.

A290163 Primes p such that A288814(4p) - A288814(3p) = 7.

A289556 Primes p such that both 5p - 4 and 4p - 5 are prime.