A097932 -id:A097932 - cp's OEIS frontend

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

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.

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.

A182138 Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 4, 0, 5, 3, 4, 2, 7, 3, 8, 6, 0, 7, 5, 1, 10, 6, 0, 9, 3, 8, 4, 2, 13, 3, 14, 12, 6, 0, 13, 11, 5, 1, 12, 0, 17, 9, 3, 16, 10, 8, 2, 19, 15, 9, 20, 18, 6, 0, 19, 17, 13, 7, 5, 22, 18, 12, 6, 21, 15, 3, 20, 16, 14, 10, 4, 25, 15, 9, 24, 18, 12, 0, 23, 17, 13, 11, 7, 1

Offset: 2

Jean COHEN, Apr 16 2012

The Goldbach conjecture is that for any even integer 2n>=4, at least one pair of primes p and q exist such that p+q=2n. The present numbers listed here are the distances d between each prime and n, the half of the even integer 2n: d=n-p=q-n with p <= q.
See the link section for plots I added. - Jason Kimberley, Oct 04 2012
Each nonzero entry d of row n is coprime to n. For otherwise n+d would be composite. - Jason Kimberley, Oct 10 2012

			n=2, 2n=4, 4=2+2, p=q=2 -> d=0.
n=18, 2n=36, four prime pairs have a sum of 36: 5+31, 7+29, 13+23, 17+19, with the four distances d being 13=18-5=31-18, 11=18-7=29-18, 5=18-13=23-18, 1=18-17=19-18.
Triangle begins:
  0;
  0;
  1;
  2, 0;
  1;
  4, 0;
  5, 3;
  4, 2;
  7, 3;
  8, 6, 0;

Alois P. Heinz, Rows n = 2..600, flattened
OEIS (Plot 2), Plot of (n, d)
Subplots for fixed p:
OEIS (Plot 2), A067076 vs A098090 (p=3).
OEIS (Plot 2), A089038 vs A089253 (p=5).
OEIS (Plot 2), A105760 vs A089192 (p=7).
...
OEIS (Plot 2), A153143 vs A097932 (p=19).
Wikipedia, Goldbach's conjecture

Cf. A045917 (row lengths), A047949 (first column), A047160 (last elements of rows).

Cf. A184995.

A182138:= func;
&cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012

T:= n-> seq(`if`(isprime(p) and isprime(2*n-p), n-p, NULL), p=2..n):
seq(T(n), n=2..40); # Alois P. Heinz, Apr 16 2012

T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], n-p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

for(n=2,18,forprime(p=2,n,if(isprime(2*n-p),print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012

T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012

A153041 Numbers n >=10 such that 2*n-19 is not a prime.

Original entry on oeis.org

10, 14, 17, 20, 22, 23, 26, 27, 29, 32, 34, 35, 37, 38, 41, 42, 44, 47, 48, 50, 52, 53, 55, 56, 57, 59, 62, 65, 67, 68, 69, 70, 71, 72, 74, 76, 77, 80, 81, 82, 83, 86, 87, 89, 90, 92, 94, 95, 97, 98, 101, 102, 103, 104

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than associated values in A153051, two more than A153047. - R. J. Mathar, Jan 05 2011

The terms after a(1) are the values of 2*h*k + k + h + 10, where h and k are positive integers.- Vincenzo Librandi, Jan 19 2013

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

Cf. A097932, A155704.
Numbers n such that 2n-k is not prime: A104275 (k=1), A153043 (k=3), A153040 (k=5), A153039 (k=7), A153044 (k=9), A153045 (k=11), A153049 (k=13), A153047 (k=15), A153051 (k=17), A153041 (k=19).
Similar sequence listed also in A089559, A153144.

[n: n in [10..150] | not IsPrime(2*n - 19)]; // Vincenzo Librandi, Jan 19 2013

Select[Range[10, 200], !PrimeQ[2 # - 19] &] (* Vincenzo Librandi, Jan 19 2013 *)

Edited by N. J. A. Sloane, Jun 22 2010

cp's OEIS Frontend

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

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A097069 Positive integers n such that 2n - 9 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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