A089253 -id:A089253 - 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

A153040 Numbers n>3 such that 2*n-5 is not a prime.

Original entry on oeis.org

7, 10, 13, 15, 16, 19, 20, 22, 25, 27, 28, 30, 31, 34, 35, 37, 40, 41, 43, 45, 46, 48, 49, 50, 52, 55, 58, 60, 61, 62, 63, 64, 65, 67, 69, 70, 73, 74, 75, 76, 79, 80, 82, 83, 85, 87, 88, 90, 91, 94, 95, 96, 97, 100, 103, 104, 105, 106, 107, 109, 110, 111

Offset: 1

Vincenzo Librandi, Dec 17 2008

One less than the associated number in A153039; one more than that in A153043. - R. J. Mathar Dec 20 2008

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

Cf. A089253, A153039.

[n: n in [4..120]|not IsPrime(2*n-5)]; // Vincenzo Librandi, Aug 30 2012

Select[Range[4,200],!PrimeQ[2*#-5]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

Let p = prime number n = (p^2+5)/2 mod (p)

Flipped sign in definition. - R. J. Mathar, Dec 20 2008

A230138 List of those primes p with p + 2 and 2*p - 5 both prime.

Original entry on oeis.org

5, 11, 17, 29, 59, 71, 101, 137, 149, 179, 197, 227, 281, 311, 431, 599, 617, 641, 809, 821, 857, 1151, 1277, 1319, 1451, 1481, 1487, 1607, 1667, 1697, 1997, 2027, 2081, 2111, 2129, 2339, 2657, 2711, 2789, 3167, 3329, 3371, 3461, 3557, 3767, 3917, 3929, 4049, 4217, 4259

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Clearly, all terms are congruent to 5 modulo 6, and not congruent to 3 modulo 5. Primes in this sequence are sparser than twin primes and Sophie Germain primes.
This sequence is interesting because of the conjectures in the comments in A230140 and A230141.
Intersection of A001359 and A089253 (or A063909). - M. F. Hasler, Oct 10 2013

			a(1) = 5 since neither 2 + 2 nor 2*3 - 5 is prime, but 5 + 2 = 7 and 2*5 - 5 = 5 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A005384, A230140, A230141.

PQ[p_]:=PQ[p]=PrimeQ[p+2]&&PrimeQ[2p-5]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,584}]

is_A230138(p)=isprime(p)&&isprime(p+2)&&isprime(p*2-5) \\ For large p it might be much faster to check first whether p%6==5. - M. F. Hasler, Oct 10 2013

A098033 Parity of p*(p+1)/2 for n-th prime p.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Jeremy Gardiner, Sep 10 2004

The following sequences (possibly with a different offset for first term) all appear to have the same parity: A034953 = triangular numbers with prime indices; A054269 = length of period of continued fraction for sqrt(p), p prime; A082749 = difference between the sum of the next prime(n) natural numbers and the sum of the next n primes; A006254 = numbers n such that 2n-1 is prime; A067076 = numbers n such that 2n+3 is a prime.
Analogous to the prime race (mod 3). - Robert G. Wilson v, Sep 17 2004
See also A089253 = 2n-5 is a prime.
For n > 1, if A000040(n) == 1 (mod 4), then a(n) = 1, otherwise a(n)=0, so (for n>1) also a(n) = number of representations of A000040(n) as a difference of hexagonal numbers (A000384) (cf. [Nyblom, p. 262]). - L. Edson Jeffery, Feb 16 2013

			a(1) = parity of (2*(2+1)/2 = 3) = 1 (odd).

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A034953, A054269, A082749, A006254, A067076, A100672.

seq((ithprime(n) mod 4) mod 3, n = 2..105] # Gary Detlefs, Oct 27 2011

Table[ Mod[ Prime[n](Prime[n] + 1)/2, 2], {n, 105}] (* Robert G. Wilson v, Sep 17 2004 *)
Mod[(#(#+1))/2,2]&/@Prime[Range[110]] (* Harvey P. Dale, Mar 29 2015 *)

a(n)=prime(n)%4<3 \\ Charles R Greathouse IV, Oct 27 2011

a(n) = parity of p*(p+1)/2 for n-th prime p.
a(n) = 1 - A100672(n), n > 1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008
For n > 1, a(n) = (prime(n) mod 4) mod 3. - Gary Detlefs, Oct 27 2011

More terms from Robert G. Wilson v, Sep 17 2004

A154681 Triangle read by rows where T(m,n) = 2mn + m + n +3.

Original entry on oeis.org

7, 10, 15, 13, 20, 27, 16, 25, 34, 43, 19, 30, 41, 52, 63, 22, 35, 48, 61, 74, 87, 25, 40, 55, 70, 85, 100, 115, 28, 45, 62, 79, 96, 113, 130, 147, 31, 50, 69, 88, 107, 126, 145, 164, 183, 34, 55, 76, 97, 118, 139, 160, 181, 202, 223, 37, 60, 83, 106, 129, 152

Offset: 1

Vincenzo Librandi, Jan 18 2009

2*T(m,n) - 5 = (2*n+1)*(2*m+1) is not prime.

First column: A112414; second column: A008587.

			Triangle begins:
  7;
  10, 15;
  13, 20, 27;
  16, 25, 34, 43;
  19, 30, 41, 52,  63;
  22, 35, 48, 61,  74,  87;
  25, 40, 55, 70,  85, 100, 115;
  28, 45, 62, 79,  96, 113, 130, 147;
  31, 50, 69, 88, 107, 126, 145, 164, 183;
  34, 55, 76, 97, 118, 139, 160, 181, 202, 223; etc.

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

Cf. A153040, A008587, A089253, A112414.

[(2*n*k + n + k + 3): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 17 2012

t[n_,k_]:=2 n*k + n + k + 3; Table[t[n, k], {n, 10}, {k, n}]//Flatten (* vincenzo Librandi, Nov 17 2012 *)

A089255 Odd numbers n such that 2*n-5 is a prime.

Original entry on oeis.org

5, 9, 11, 17, 21, 23, 29, 33, 39, 47, 51, 53, 57, 59, 71, 77, 81, 89, 93, 99, 101, 117, 119, 123, 131, 137, 141, 143, 149, 159, 161, 171, 177, 179, 189, 197, 201, 203, 207, 213, 219, 227, 231, 233, 257, 263, 273, 281, 287, 291, 299, 303, 309, 311, 323, 329, 333

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.

Cf. A089253, A002144.

Corrected sequence and extended by Vincenzo Librandi, Mar 30 2010

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A118469 Triangle read by rows: a(n,m) = If(n = 1, then 1, else Prime(n) - 1 + Sum_{k=n..m} (Prime(k + 1) - Prime(k))/2 ).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 6, 7, 8, 1, 7, 8, 9, 11, 1, 9, 10, 11, 13, 14, 1, 10, 11, 12, 14, 15, 17, 1, 12, 13, 14, 16, 17, 19, 20, 1, 15, 16, 17, 19, 20, 22, 23, 25, 1, 16, 17, 18, 20, 21, 23, 24, 26, 29, 1, 19, 20, 21, 23, 24, 26, 27, 29, 32, 33, 1, 21, 22, 23, 25, 26, 28, 29, 31, 34, 35

Offset: 1

Roger L. Bagula, May 04 2006

An improved triangular Goldbach sequence in which the gap sum is taken from a start at n.

			1
1, 3
1, 4, 5
1, 6, 7, 8
1, 7, 8, 9, 11
1, 9, 10, 11, 13, 14
1, 10, 11, 12, 14, 15, 17
1, 12, 13, 14, 16, 17, 19, 20
1, 15, 16, 17, 19, 20, 22, 23, 25
1, 16, 17, 18, 20, 21, 23, 24, 26, 29

Main diagonal: A078444, 2nd diagonal: A073273.

Columns 1-8: A000012, A006254, A098090, A089253, A097069, A097338, A097480, A098605.

t[n_, m_] := If[n == 1, 1, Prime[n] + Sum[(Prime[k + 1] - Prime[k])/2, {k, n, m}] - 1]; Table[ t[n, m], {m, 11}, {n, m}] // Flatten

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).

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A153040 Numbers n>3 such that 2*n-5 is not a prime.

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A230138 List of those primes p with p + 2 and 2*p - 5 both prime.

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PARI

A098033 Parity of p*(p+1)/2 for n-th prime p.

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A154681 Triangle read by rows where T(m,n) = 2*m*n + m + n +3.

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A089255 Odd numbers n such that 2*n-5 is a prime.

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A118469 Triangle read by rows: a(n,m) = If(n = 1, then 1, else Prime(n) - 1 + Sum_{k=n..m} (Prime(k + 1) - Prime(k))/2 ).

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A154681 Triangle read by rows where T(m,n) = 2mn + m + n +3.