A089192 -id:A089192 - cp's OEIS frontend

A153039 Numbers k such that 2*k-7 is composite.

8, 11, 14, 16, 17, 20, 21, 23, 26, 28, 29, 31, 32, 35, 36, 38, 41, 42, 44, 46, 47, 49, 50, 51, 53, 56, 59, 61, 62, 63, 64, 65, 66, 68, 70, 71, 74, 75, 76, 77, 80, 81, 83, 84, 86, 88, 89, 91, 92, 95, 96, 97, 98, 101, 104, 105, 106, 107, 108, 110, 111, 112

Offset: 1

Vincenzo Librandi, Dec 17 2008

Two more than the associated value in A153043, one more than in A153040.

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

Complement of A089192, A153040.

[n: n in [5..220]| not IsPrime(2*n-7)];

Select[Range[8,200], !PrimeQ[2*#-7]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

list(n)=apply(k->(k+7)/2,setminus(vector(n,k,2*k+7), primes(primepi(2*n+7)))) \\ Charles R Greathouse IV, May 14 2012

Partially edited by N. J. A. Sloane, Jun 23 2010

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

A088767 a(n) = A087697(n)/2.

Original entry on oeis.org

5, 6, 12, 15, 18, 27, 30, 33, 45, 48, 60, 72, 78, 87, 93, 102, 117, 132, 135, 138, 150, 162, 180, 183, 195, 213, 225, 228, 258, 282, 285, 297, 300, 303, 312, 327, 333, 342, 363, 375, 390, 402, 408, 423, 435, 480, 492, 495, 513, 528, 555, 558, 597, 612, 615, 642

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Numbers n such that 2*n-7 [A089192] and 2*n+7 [A105760] are both prime. [Vincenzo Librandi, Jul 10 2010]

Cf. A087697, A040040, A088763, A088765, A088769.

Cf. A089192, A105760. [Vincenzo Librandi, Jan 18 2009]

A063910 Primes p such that 2*p - 7 is also prime.

Original entry on oeis.org

5, 7, 13, 19, 37, 43, 67, 73, 79, 103, 109, 139, 157, 193, 223, 277, 307, 313, 349, 367, 373, 379, 397, 409, 433, 457, 463, 487, 499, 523, 547, 619, 643, 727, 733, 739, 769, 787, 853, 877, 883, 919, 937, 997, 1009, 1069, 1093, 1123, 1129, 1237

Offset: 1

N. J. A. Sloane, Aug 31 2001

All terms > 5 are == 1 (mod 6). - Zak Seidov, Jan 07 2014

Harry J. Smith, Table of n, a(n) for n = 1..1000

[n: n in [3..2000] | IsPrime(n) and IsPrime(2*n-7)]; // Vincenzo Librandi, Feb 02 2014

Select[Prime[Range[2, 2000]], PrimeQ[2 # - 7]&] (* Vincenzo Librandi, Feb 02 2014 *)

isok(p) = { isprime(p) && isprime(2*p - 7) } \\ Harry J. Smith, Sep 02 2009

A089192 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

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A153039 Numbers k such that 2*k-7 is composite.

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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).

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A088767 a(n) = A087697(n)/2.

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A063910 Primes p such that 2*p - 7 is also prime.

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