A089192 -id:A089192 - cp's OEIS frontend

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

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.

A097932 Positive integers n such that 2n-19 is prime.

Original entry on oeis.org

11, 12, 13, 15, 16, 18, 19, 21, 24, 25, 28, 30, 31, 33, 36, 39, 40, 43, 45, 46, 49, 51, 54, 58, 60, 61, 63, 64, 66, 73, 75, 78, 79, 84, 85, 88, 91, 93, 96, 99, 100, 105, 106, 108, 109, 115, 121, 123, 124, 126, 129, 130, 135, 138, 141, 144, 145, 148, 150, 151, 156, 163

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 21 2004

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

Cf. A000040.
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), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), this sequence (k=19).

(Prime[Range[2,100]]+19)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[10,200],PrimeQ[2#-19]&] (* Harvey P. Dale, May 08 2017 *)

Half of p+19 where p is a prime greater than 2.

A153081 Nonnegative numbers k such that 2k + 13 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 12, 14, 15, 17, 20, 23, 24, 27, 29, 30, 33, 35, 38, 42, 44, 45, 47, 48, 50, 57, 59, 62, 63, 68, 69, 72, 75, 77, 80, 83, 84, 89, 90, 92, 93, 99, 105, 107, 108, 110, 113, 114, 119, 122, 125, 128, 129, 132, 134, 135, 140, 147, 149, 150, 152, 159, 162, 167

Offset: 1

Vincenzo Librandi, Dec 18 2008

Or, (p-13)/2 for primes p >= 13.
a(n) = (A000040(n+5) - 13)/2.
a(n) = A005097(n+4) - 6.
a(n) = A067076(n+4) - 5.
a(n) = A089038(n+3) - 4.
a(n) = A105760(n+2) - 3.
a(n) = A101448(n+1) - 1.
a(n) = A089559(n-1) + 1 for n > 1.

			For k = 7, 2*k+13 = 27 is not prime, so 7 is not in the sequence;
for k = 8, 2*k+13 = 29 is prime, so 8 is in the sequence.

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

Cf. A000040 (prime numbers).
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), this seq (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+13) ); # G. C. Greubel, May 22 2019

[ n: n in [0..200] | IsPrime(2*n+13) ];

(Prime[Range[6,100]]-13)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0,200],PrimeQ[2#+13]&] (* Harvey P. Dale, Mar 02 2015 *)

is(n)=isprime(2*n+13) \\ Charles R Greathouse IV, Jul 12 2016

[n for n in (0..200) if is_prime(2*n+13) ] # G. C. Greubel, May 22 2019

Edited and extended by Klaus Brockhaus, Dec 22 2008

Definition clarified by Zak Seidov, Jul 11 2014

A097338 Positive integers n such that 2n-11 is prime.

Original entry on oeis.org

7, 8, 9, 11, 12, 14, 15, 17, 20, 21, 24, 26, 27, 29, 32, 35, 36, 39, 41, 42, 45, 47, 50, 54, 56, 57, 59, 60, 62, 69, 71, 74, 75, 80, 81, 84, 87, 89, 92, 95, 96, 101, 102, 104, 105, 111, 117, 119, 120, 122, 125, 126, 131, 134, 137, 140, 141, 144, 146, 147, 152, 159, 161

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 17 2004

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

Cf. A000040.
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), A097069 (k=9), this sequence (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+11)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[6,200],PrimeQ[2#-11]&] (* Harvey P. Dale, May 24 2014 *)

is(n)=isprime(2*n-11) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+11 where p is a prime greater than 2.

A097480 Positive integers n such that 2n-15 is prime.

Original entry on oeis.org

9, 10, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 34, 37, 38, 41, 43, 44, 47, 49, 52, 56, 58, 59, 61, 62, 64, 71, 73, 76, 77, 82, 83, 86, 89, 91, 94, 97, 98, 103, 104, 106, 107, 113, 119, 121, 122, 124, 127, 128, 133, 136, 139, 142, 143, 146, 148, 149, 154, 161, 163

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 19 2004

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

Cf. A000040, A098602, A098603.
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), A097069 (k=9), A097338 (k=11), A097363 (k=13), this sequence (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+15)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[9,200],PrimeQ[2#-15]&] (* Harvey P. Dale, Apr 04 2021 *)

is(n)=isprime(2*n-15) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+15 where p is a prime greater than 2.

A098605 Positive integers n such that 2n - 17 is prime.

Original entry on oeis.org

10, 11, 12, 14, 15, 17, 18, 20, 23, 24, 27, 29, 30, 32, 35, 38, 39, 42, 44, 45, 48, 50, 53, 57, 59, 60, 62, 63, 65, 72, 74, 77, 78, 83, 84, 87, 90, 92, 95, 98, 99, 104, 105, 107, 108, 114, 120, 122, 123, 125, 128, 129, 134, 137, 140, 143, 144, 147, 149, 150, 155, 162

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 20 2004

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

Cf. A000040, A098602, A098603.
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), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), this sequence (k=17), A097932 (k=19).

(Prime[Range[2,100]]+17)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)

is(n)=isprime(2*n-17) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+17 where p is a prime greater than 2.

A173059 Nonnegative numbers k such that 2*k + 17 is prime.

Original entry on oeis.org

0, 1, 3, 6, 7, 10, 12, 13, 15, 18, 21, 22, 25, 27, 28, 31, 33, 36, 40, 42, 43, 45, 46, 48, 55, 57, 60, 61, 66, 67, 70, 73, 75, 78, 81, 82, 87, 88, 90, 91, 97, 103, 105, 106, 108, 111, 112, 117, 120, 123, 126, 127, 130, 132, 133, 138, 145, 147, 148, 150, 157, 160, 165

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

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), this seq (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+17) ); # G. C. Greubel, May 22 2019

[n: n in [0..200] | IsPrime(2*n+17) ]; // G. C. Greubel, May 22 2019

Mathematica
```
(Prime[Range[7,100]]-17)/2
```

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

[n for n in (0..200) if is_prime(2*n+17) ] # G. C. Greubel, May 22 2019

Definition clarified by Zak Seidov, Jul 11 2014

A097363 Positive integers n such that 2n-13 is prime.

Original entry on oeis.org

8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 25, 27, 28, 30, 33, 36, 37, 40, 42, 43, 46, 48, 51, 55, 57, 58, 60, 61, 63, 70, 72, 75, 76, 81, 82, 85, 88, 90, 93, 96, 97, 102, 103, 105, 106, 112, 118, 120, 121, 123, 126, 127, 132, 135, 138, 141, 142, 145, 147, 148, 153, 160, 162

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 18 2004

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

Cf. A000040, A098602.
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), A097069 (k=9), A097338 (k=11), this sequence (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+13)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[8,200],PrimeQ[2#-13]&] (* Harvey P. Dale, Apr 26 2013 *)

Half of p+13 where p is a prime greater than 2.

A081759 Numbers n such that 5n+6 is prime.

Original entry on oeis.org

1, 5, 7, 11, 13, 19, 25, 29, 35, 37, 41, 47, 49, 53, 55, 61, 65, 79, 83, 85, 91, 97, 103, 107, 113, 119, 125, 127, 131, 137, 139, 149, 151, 161, 163, 175, 181, 187, 193, 197, 203, 205, 209, 211, 217, 229, 233, 235, 239, 245, 257, 259, 263, 271, 275, 289

Offset: 1

Giovanni Teofilatto, Nov 21 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

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A024912, A030430, A049511, A067076, A087505, A089192, A089253.

[n: n in [0..300]| IsPrime(5*n + 6)]; // Vincenzo Librandi, Oct 16 2012

A081759 := proc(n) option remember: local k: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(isprime(5*k+6))then return k: fi: od: end: seq(A081759(n),n=1..100); # Nathaniel Johnston, May 28 2011

Select[Range[300], PrimeQ[5# + 6] &] (* Ray Chandler, Dec 06 2006 *)

is(n)=isprime(5*n+6) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = 2*A024912(n) - 1.

Corrected by Ray Chandler, Nov 22 2003

A154685 Triangle read by rows: T(n, k) = 2nk + n + k + 4.

Original entry on oeis.org

8, 11, 16, 14, 21, 28, 17, 26, 35, 44, 20, 31, 42, 53, 64, 23, 36, 49, 62, 75, 88, 26, 41, 56, 71, 86, 101, 116, 29, 46, 63, 80, 97, 114, 131, 148, 32, 51, 70, 89, 108, 127, 146, 165, 184, 35, 56, 77, 98, 119, 140, 161, 182, 203, 224, 38, 61, 84, 107, 130, 153, 176, 199, 222, 245, 268

Offset: 1

Vincenzo Librandi, Jan 18 2009

The terms form a subset of A153039 because 2*T(n, k) - 7 = (2*n+1)*(2*k+1) are not prime.

			Triangle begins:
   8;
  11, 16;
  14, 21, 28;
  17, 26, 35, 44;
  20, 31, 42, 53,  64;
  23, 36, 49, 62,  75,  88;
  26, 41, 56, 71,  86, 101, 116;
  29, 46, 63, 80,  97, 114, 131, 148;
  32, 51, 70, 89, 108, 127, 146, 165, 184;
  35, 56, 77, 98, 119, 140, 161, 182, 203, 224;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A089192, A153039.
Cf. A151675 (row sums).
Similar triangle: A155724.
Columns k: A016789 (k=1), A016861 (k=2).
Main diagonal: A137882, A271649.

A154685:= func< n,k | 2*n*k+n+k+4 >;
[A154685(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 21 2025

Flatten@Table[2*n*m+m+n+4,{n,20},{m,n}] (* Vincenzo Librandi, Jan 29 2012 *)

for(m=1,9,for(n=1,m,print1(2*m*n+m+n+4", "))) \\ Charles R Greathouse IV, Dec 27 2011

def A154685(n,k): return 2*n*k+n+k+4
print(flatten([[A154685(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

Sum_{k=1..n} T(n, k) = A151675(n). - N. J. A. Sloane, May 31 2009
T(n, k) = A155724(n,k) + 8. - L. Edson Jeffery, Oct 12 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n + 3.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*(9*(1-(-1)^n) + 2*(2-3*(-1)^n)*n - 4*(-1)^n*n^2).
G.f.: x*y*(8 - 5*(x+y) + 4*x*y)/((1-x)*(1-y))^2.
E.g.f.: 4 - (4+x)*exp(x) - (4+y)*exp(y) + (4+x+y+2*x*y)*exp(x+y).
 (End)

Clarified comment. - R. J. Mathar, Jan 24 2009

cp's OEIS Frontend

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

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A097932 Positive integers n such that 2n-19 is prime.

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A153081 Nonnegative numbers k such that 2k + 13 is prime.

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A097338 Positive integers n such that 2n-11 is prime.

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A097480 Positive integers n such that 2n-15 is prime.

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A098605 Positive integers n such that 2n - 17 is prime.

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A173059 Nonnegative numbers k such that 2*k + 17 is prime.

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GAP

Magma

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Sage

Extensions

A097363 Positive integers n such that 2n-13 is prime.

A154685 Triangle read by rows: T(n, k) = 2nk + n + k + 4.