A067076 -id:A067076 - cp's OEIS frontend

A144562 Triangle read by rows: T(n, k) = 2nk + n + k - 1.

3, 6, 11, 9, 16, 23, 12, 21, 30, 39, 15, 26, 37, 48, 59, 18, 31, 44, 57, 70, 83, 21, 36, 51, 66, 81, 96, 111, 24, 41, 58, 75, 92, 109, 126, 143, 27, 46, 65, 84, 103, 122, 141, 160, 179, 30, 51, 72, 93, 114, 135, 156, 177, 198, 219, 33, 56, 79, 102, 125, 148, 171, 194, 217, 240, 263

Offset: 1

Vincenzo Librandi, Jan 06 2009

Rearrangement of A153238, numbers n such that 2*n+3 is not prime (we have 2*T(n,k) + 3 = (2*n+1)*(2*k+1), as 2*n+3 is odd it consists of (at least) two odd factors and all such factors appear by definition).

			Triangle begins:
   3;
   6, 11;
   9, 16, 23;
  12, 21, 30, 39;
  15, 26, 37, 48,  59;
  18, 31, 44, 57,  70,  83;
  21, 36, 51, 66,  81,  96, 111;
  24, 41, 58, 75,  92, 109, 126, 143;
  27, 46, 65, 84, 103, 122, 141, 160, 179;
  ...

Vincenzo Librandi, Rows n = 1..100, flattened
Mutsumi Suzuki Vincenzo Librandi's method for sequential primes (Librandi's description in Italian).

Cf. A067076, A144640, A153238.
Main diagonal gives A142463.
T(2n,n) gives A180863(n+1).

[2*n*k+n+k-1: k in [1..n], n in [1..11]]; /* or, see example: */ [[2*n*k+n+k-1: k in [1..n]]: n in [1..9]]; // Bruno Berselli, Dec 04 2011

A144562:= (n,k) -> 2*n*k +n +k -1; seq(seq(A144562(n,k), k=1..n), n=1..12); # G. C. Greubel, Mar 01 2021

T[n_, k_]:= 2*n*k +n +k -1; Table[T[n, k], {n, 11}, {k, n}]//Flatten

T(n,k)=2*n*k+n+k-1 \\ Charles R Greathouse IV, Dec 28 2011

flatten([[2*n*k+n+n-1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 01 2021

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n - 1)/2 = A144640(n). - G. C. Greubel, Mar 01 2021

G.f.: x*y*(3 + 2*x*y + 2*x^3*y^2 - x^2*y*(6 + y))/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Nov 04 2024

Edited by Ray Chandler, Jan 07 2009

A023204 Primes p such that 2*p + 3 is also prime.

Original entry on oeis.org

2, 5, 7, 13, 17, 19, 29, 43, 47, 53, 67, 73, 89, 97, 113, 127, 137, 139, 157, 167, 173, 193, 197, 199, 223, 227, 229, 269, 277, 283, 307, 337, 349, 353, 379, 383, 397, 409, 439, 463, 467, 487, 503, 509, 523, 547, 557, 563, 599, 607, 613, 617, 643, 647, 659, 739, 743, 773

Offset: 1

David W. Wilson

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

Cf. A089530, A089531, A089532.

[p: p in PrimesUpTo(1000) | IsPrime(2*p+3)]; // Vincenzo Librandi, Nov 20 2010

A023204 := proc (n) if isprime(n) and isprime(2*n+3) = true then n end if end proc; seq(A023204(n), n=1..1000); # Wesley Ivan Hurt, Jun 28 2014

f[n_]:=PrimeQ[2*n+3]; lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
Select[Prime[Range[200]],PrimeQ[2#+3]&] (* Harvey P. Dale, May 03 2012 *)

is(n)=isprime(2*n+3)&&isprime(n) \\ Charles R Greathouse IV, Jul 02 2013

A067076 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

A105760 Nonnegative numbers k such that 2k+7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 11, 12, 15, 17, 18, 20, 23, 26, 27, 30, 32, 33, 36, 38, 41, 45, 47, 48, 50, 51, 53, 60, 62, 65, 66, 71, 72, 75, 78, 80, 83, 86, 87, 92, 93, 95, 96, 102, 108, 110, 111, 113, 116, 117, 122, 125, 128, 131, 132, 135, 137, 138, 143, 150, 152, 153, 155, 162

Offset: 1

Parthasarathy Nambi, May 04 2005

			If n=0, then 2*0 + 7 = 7 (prime).
If n=15, then 2*15 + 7 = 37 (prime).
If n=27, then 2*27 + 7 = 61 (prime).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

Cf. A153053 (Numbers n such that 2n+7 is not a prime)
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), this seq(k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+7) ); # G. C. Greubel, May 21 2019

[n: n in [0..200]| IsPrime(2*n+7)]; // Vincenzo Librandi, Dec 21 2010

(Prime[Range[4,100]]-7)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 200], PrimeQ[2 # + 7] &] (* Vincenzo Librandi, May 20 2014 *)

is(n)=isprime(2*n+7) \\ Charles R Greathouse IV, Feb 16 2017

[n for n in (0..200) if is_prime(2*n+7) ] # G. C. Greubel, May 21 2019

More terms from Rick L. Shepherd, May 18 2005

A089253 Numbers n such that 2n - 5 is a prime.

Original entry on oeis.org

4, 5, 6, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 26, 29, 32, 33, 36, 38, 39, 42, 44, 47, 51, 53, 54, 56, 57, 59, 66, 68, 71, 72, 77, 78, 81, 84, 86, 89, 92, 93, 98, 99, 101, 102, 108, 114, 116, 117, 119, 122, 123, 128, 131, 134, 137, 138, 141, 143, 144, 149, 156, 158, 159, 161

Offset: 1

Giovanni Teofilatto, Dec 12 2003

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

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).

Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), this sequence (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+5)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[2,200],PrimeQ[2#-5]&] (* Harvey P. Dale, Dec 29 2024 *)

a(n)=prime(n+1)\2+3 \\ Charles R Greathouse IV, Oct 30 2012

a(n) = A067076(n) + 4. - Giovanni Teofilatto, Dec 14 2003

Corrected by Ralf Stephan, Mar 03 2004

Further correction from Jeremy Gardiner, Sep 11 2004

A089192 Numbers n such that 2n - 7 is a prime.

Original entry on oeis.org

5, 6, 7, 9, 10, 12, 13, 15, 18, 19, 22, 24, 25, 27, 30, 33, 34, 37, 39, 40, 43, 45, 48, 52, 54, 55, 57, 58, 60, 67, 69, 72, 73, 78, 79, 82, 85, 87, 90, 93, 94, 99, 100, 102, 103, 109, 115, 117, 118, 120, 123, 124, 129, 132, 135, 138, 139, 142, 144, 145, 150, 157, 159, 160

Offset: 1

Giovanni Teofilatto, Dec 08 2003

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

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).

Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), this sequence (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Select[Range[3,200],PrimeQ[2#-7]&] (* Harvey P. Dale, Aug 24 2014 *)

is(n)=isprime(2*n - 7) \\ Charles R Greathouse IV, Apr 28 2015

Corrected by Ralf Stephan, Mar 03 2004

Further correction from Jeremy Gardiner, Sep 11 2004

A153143 Nonnegative numbers k such that 2k + 19 is prime.

Original entry on oeis.org

0, 2, 5, 6, 9, 11, 12, 14, 17, 20, 21, 24, 26, 27, 30, 32, 35, 39, 41, 42, 44, 45, 47, 54, 56, 59, 60, 65, 66, 69, 72, 74, 77, 80, 81, 86, 87, 89, 90, 96, 102, 104, 105, 107, 110, 111, 116, 119, 122, 125, 126, 129, 131, 132, 137, 144, 146, 147, 149, 156, 159, 164, 165

Offset: 1

Vincenzo Librandi, Dec 19 2008

Or, (p-19)/2 for primes p >= 19.
a(n) = (A000040(n+7) - 19)/2.
a(n) = A005097(n+6) - 9.
a(n) = A067076(n+6) - 8.
a(n) = A089038(n+5) - 7.
a(n) = A105760(n+4) - 6.
a(n) = A101448(n+3) - 4.
a(n) = A089559(n+1) - 2.

			For k = 4, 2*k+19 = 27 is not prime, so 4 is not in the sequence;
for k = 17, 2*k+19 = 53 is prime, so 17 is in the sequence.

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

Cf. A000040 (prime numbers), A153144 (2n+19 is not prime).
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), this seq (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+19) ); # G. C. Greubel, May 22 2019

[ n: n in [0..165] | IsPrime(2*n+19) ];

(Prime[Range[8,100]]-19)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 170], PrimeQ[(2*#)+19]&] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(2*n+19) \\ Charles R Greathouse IV, Feb 17 2017

list(lim)=apply(p->p\2-9, primes([19,lim\1*2+19])) \\ Charles R Greathouse IV, Jan 03 2025

[n for n in (0..200) if is_prime(2*n+19) ] # G. C. Greubel, May 22 2019

a(n) ~ (n/2) log n. - Charles R Greathouse IV, Jan 03 2025

Edited, corrected and extended by Klaus Brockhaus, Dec 22 2008

Definition clarified by Zak Seidov, Jul 11 2014

A155722 Numbers k such that 2*k + 9 is prime.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 22, 25, 26, 29, 31, 32, 35, 37, 40, 44, 46, 47, 49, 50, 52, 59, 61, 64, 65, 70, 71, 74, 77, 79, 82, 85, 86, 91, 92, 94, 95, 101, 107, 109, 110, 112, 115, 116, 121, 124, 127, 130, 131, 134, 136, 137, 142, 149, 151, 152, 154, 161, 164

Offset: 1

Vincenzo Librandi, Jan 25 2009

Subsequence of A001651; A011655(a(n)) = 1. - Reinhard Zumkeller, Jul 09 2010

One less than the associated entry in A105760, two less than in A089038, three less than in A067076. - R. J. Mathar, Jan 05 2011

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

Cf. A155723, A155724.

Numbers h such that 2*h + k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), this seq (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).

Filtered([0..200], k-> IsPrime(2*k+9)); # G. C. Greubel, May 21 2019

[n: n in [0..200] | IsPrime(2*n+9)]; // Vincenzo Librandi, Sep 24 2012

(Prime[Range[5, 100]] - 9)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 200], PrimeQ[2 # + 9]&] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(2*n+9) \\ Charles R Greathouse IV, Jul 12 2016

[n for n in (0..200) if is_prime(2*n+9) ] # G. C. Greubel, May 21 2019

Edited by N. J. A. Sloane, Jun 23 2010

Definition clarified by Zak Seidov, Jul 11 2014

A089559 Nonnegative numbers n such that 2*n + 15 is prime.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 23, 26, 28, 29, 32, 34, 37, 41, 43, 44, 46, 47, 49, 56, 58, 61, 62, 67, 68, 71, 74, 76, 79, 82, 83, 88, 89, 91, 92, 98, 104, 106, 107, 109, 112, 113, 118, 121, 124, 127, 128, 131, 133, 134, 139, 146, 148, 149, 151, 158, 161, 166

Offset: 1

Ray Chandler, Nov 29 2003

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

Cf. A086303.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), this seq (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+15) ); # G. C. Greubel, May 22 2019

[n: n in [0..200] |  IsPrime(2*n+15)]; // Vincenzo Librandi, Apr 28 2014

(Prime[Range[7,100]]-15)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 200], PrimeQ[2 # + 15] &] (* Vincenzo Librandi, Apr 28 2014 *)

is(n)=isprime(2*n+15) \\ Charles R Greathouse IV, Apr 28 2015

[n for n in (0..200) if is_prime(2*n+15) ] # G. C. Greubel, May 22 2019

a(n) = A086303(n)/2.

Definition clarified by Zak Seidov, Jul 11 2014

A101448 Nonnegative numbers k such that 2k + 11 is prime.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 13, 15, 16, 18, 21, 24, 25, 28, 30, 31, 34, 36, 39, 43, 45, 46, 48, 49, 51, 58, 60, 63, 64, 69, 70, 73, 76, 78, 81, 84, 85, 90, 91, 93, 94, 100, 106, 108, 109, 111, 114, 115, 120, 123, 126, 129, 130, 133, 135, 136, 141, 148, 150, 151, 153, 160, 163

Offset: 1

Parthasarathy Nambi, Jan 24 2005

2 is the smallest single-digit prime and 11 is the smallest two-digit prime.

			If n=1, then 2*1 + 11 = 13 (prime).
If n=49, then 2*49 + 11 = 109 (prime).
If n=69, then 2*69 + 11 = 149 (prime).

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101123, A101086.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), this seq (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+11) ); # G. C. Greubel, May 21 2019

[n: n in [0..200] | IsPrime(2*n+11)]; // Vincenzo Librandi, Nov 17 2010

select(k-> isprime(11+2*k), [$0..200])[];  # Alois P. Heinz, Jun 02 2022

Select[Range[0, 200], PrimeQ[2# + 11] &] (* Stefan Steinerberger, Feb 28 2006 *)

is(n)=isprime(2*n+11) \\ Charles R Greathouse IV, Apr 29 2015

[n for n in (0..200) if is_prime(2*n+11) ] # G. C. Greubel, May 21 2019

More terms from Stefan Steinerberger, Feb 28 2006

Definition clarified by Zak Seidov, Jul 11 2014

A097069 Positive integers n such that 2n - 9 is prime.

Original entry on oeis.org

6, 7, 8, 10, 11, 13, 14, 16, 19, 20, 23, 25, 26, 28, 31, 34, 35, 38, 40, 41, 44, 46, 49, 53, 55, 56, 58, 59, 61, 68, 70, 73, 74, 79, 80, 83, 86, 88, 91, 94, 95, 100, 101, 103, 104, 110, 116, 118, 119, 121, 124, 125, 130, 133, 136, 139, 140, 143, 145, 146, 151, 158, 160

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 15 2004

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

Cf. A000040, A086801.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), this seq(k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

[n: n in [6..160] | IsPrime(2*n-9)];  // Bruno Berselli, Mar 05 2011

(Prime[Range[2,100]]+9)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[4, 200], PrimeQ[2 # - 9] &] (* Vincenzo Librandi, Oct 16 2012 *)

is(n)=isprime(2*n-9) \\ Charles R Greathouse IV, Apr 28 2015

Half of p+9 where p is a prime greater than 2.

cp's OEIS Frontend

A144562 Triangle read by rows: T(n, k) = 2*n*k + n + k - 1.

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A023204 Primes p such that 2*p + 3 is also prime.

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A105760 Nonnegative numbers k such that 2k+7 is prime.

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A089253 Numbers n such that 2n - 5 is a prime.

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A089192 Numbers n such that 2n - 7 is a prime.

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A153143 Nonnegative numbers k such that 2k + 19 is prime.

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A144562 Triangle read by rows: T(n, k) = 2nk + n + k - 1.