A153143 -id:A153143 - cp's OEIS frontend

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).

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

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

Original entry on oeis.org

-5, -2, 3, 1, 8, 15, 4, 13, 22, 31, 7, 18, 29, 40, 51, 10, 23, 36, 49, 62, 75, 13, 28, 43, 58, 73, 88, 103, 16, 33, 50, 67, 84, 101, 118, 135, 19, 38, 57, 76, 95, 114, 133, 152, 171, 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 25, 48, 71, 94, 117, 140, 163, 186, 209

Offset: 1

Vincenzo Librandi, Jan 24 2009

The numbers 2*T(m,n)+19 =(2*n+1)*(2*m+1) are not prime.

First column: A016777, second column: A016885, third column: A016993, fourth column: A017209. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
-5;
-2, 3;
1,  8,  15;
4,  13, 22, 31;
7,  18, 29, 40, 51;
10, 23, 36, 49, 62,  75;
13, 28, 43, 58, 73,  88,  103;
16, 33, 50, 67, 84,  101, 118, 135;
19, 38, 57, 76, 95,  114, 133, 152, 171;
22, 43, 64, 85, 106, 127, 148, 169, 190, 211; etc.

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

Cf. A153143, A153144, A016777, A016885, A016993, A017209.

[2*n*k + n + k - 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k - 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A153145 Primes p such that 2*p + 19 is also prime.

Original entry on oeis.org

2, 5, 11, 17, 41, 47, 59, 89, 107, 131, 137, 149, 167, 191, 251, 269, 311, 317, 389, 401, 419, 431, 461, 467, 479, 521, 587, 599, 641, 677, 797, 809, 839, 857, 929, 941, 947, 977, 1031, 1061, 1097, 1109, 1181, 1187, 1229, 1301, 1307, 1319, 1361, 1367, 1409

Offset: 1

Vincenzo Librandi, Dec 19 2008

			For n=2, 2*n+19 = 23 is prime, so 2 is in the sequence.

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

Cf. A153143 (m and 2*m+19 are both prime), A005384 (Sophie Germain primes, m and 2*m+1 are both prime), A023204 (m and 2*m+3 are both prime), A023205 (m and 2*m+5 are both prime), A023206 (m and 2*m+7 are both prime), A023207 (m and 2*m+9 are both prime).

[p: p in PrimesUpTo(1500) | IsPrime(2*p+19)];

Select[Prime[Range[2000]],PrimeQ[2 # + 19] &] (* Vincenzo Librandi, Oct 20 2012 *)

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

A153146 Numbers n such that 2n + 19 and 2n - 19 are prime.

Original entry on oeis.org

11, 12, 21, 24, 30, 39, 45, 54, 60, 66, 96, 105, 126, 129, 144, 156, 165, 189, 201, 210, 219, 234, 240, 261, 264, 291, 294, 306, 336, 360, 369, 396, 420, 429, 450, 474, 486, 495, 501, 516, 525, 534, 555, 591, 606, 639, 651, 654, 690, 714, 726, 735, 756, 765

Offset: 1

Vincenzo Librandi, Dec 19 2008

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

Cf. A153041, A097932, A153143.

[n: n in [7..900] | IsPrime(2*n - 19) and IsPrime(2*n + 19)]; // Vincenzo Librandi, Oct 20 2012

Select[Range[7, 4000], PrimeQ[2 # - 19] && PrimeQ[2 # + 19] &] (* Vincenzo Librandi, Oct 20 2012 *)

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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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A155551 Triangle read by rows where T(m,n)=2*m*n + m + n - 9.

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A153145 Primes p such that 2*p + 19 is also prime.

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A153146 Numbers n such that 2*n + 19 and 2*n - 19 are prime.

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Magma

Mathematica

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

A153146 Numbers n such that 2n + 19 and 2n - 19 are prime.