A154720 -id:A154720 - cp's OEIS frontend

A184995 Irregular triangle T, read by rows, in which row n lists the primes p <= n such that 2n-p is also prime.

2, 3, 3, 3, 5, 5, 3, 7, 3, 5, 5, 7, 3, 7, 3, 5, 11, 5, 7, 11, 3, 7, 13, 5, 11, 7, 11, 13, 3, 13, 3, 5, 11, 17, 5, 7, 13, 17, 7, 19, 3, 11, 17, 5, 11, 13, 19, 3, 7, 13, 3, 5, 17, 23, 5, 7, 11, 17, 19, 3, 7, 13, 19, 5, 11, 23, 7, 11, 13, 17, 23, 3, 13, 19, 5, 11, 17, 29, 7, 13, 17, 19, 23, 29

Offset: 2

Jason Kimberley, Sep 03 2011

Row n has first entry A020481(n), length A045917(n), and last entry A112823(n).
Each row is the prefix to the middle of the corresponding row of A171637.
The Goldbach conjecture states that this irregular Goldbach triangle has in each row at least one entry (A045917(n) >= 1). - Wolfdieter Lang, May 14 2016

			The irregular triangle T(n, i) starts:
n, 2*n\i  1   2   3   4   5   6 ...
2,   4    2
3,   6    3
4,   8    3
5,  10    3   5
6,  12    5
7,  14    3   7
8,  16    3   5
9,  18    5   7
10, 20    3   7
11, 22    3   5  11
12, 24    5   7  11
13, 26    3   7  13
14, 28    5  11
15, 30    7  11  13
16, 32    3  13
17, 34    3   5  11  17
18, 36    5   7  13  17
19, 38    7  19
20, 40    3  11  17
21, 42    5  11  13  19
22, 44    3   7  13
23, 46    3   5  17  23
24, 48    5   7  11  17  19
25, 50    3   7  13  19
26, 52    5  11  23
27, 54    7  11  13  17  23
28, 56    3  13  19
29, 58    5  11  17  29
30, 60    7  13  17  19  23  29
... reformatted - _Wolfdieter Lang_, May 14 2016

Jason Kimberley, Table of n, a(n) for n = 2..1000 (flattened 2..26552)
OEIS (Plot 2), Plot of (n,p)
Index entries for sequences related to Goldbach conjecture

Related triangles: A154720, A154721, A154722, A154723, A154724, A154725, A154726, A154727, A171637, A182138.

A184995 := func;
&cat[A184995(n):n in [2..30]];

T:= n-> seq(`if`(andmap(isprime, [p, 2*n-p]), p, NULL), p=2..n):
seq(T(n), n=2..40);  # Alois P. Heinz, Jan 09 2025

Table[Select[Prime@ Range@ PrimePi@ n, PrimeQ[2 n - #] &], {n, 2, 30}] // Flatten (* Michael De Vlieger, May 14 2016 *)
T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz in A182138 *)

T(n,i) = n - A182138(n,i). - Jason Kimberley, Sep 25 2012

A154784 Row sums of triangle in A154721.

Original entry on oeis.org

0, 4, 6, 16, 10, 24, 28, 32, 54, 60, 44, 96, 52, 56, 120, 96, 102, 144, 76, 120, 210, 176, 138, 288, 200, 156, 324, 168, 174, 420, 186, 320, 396, 204, 350, 504, 370, 380, 546, 400, 328, 756, 344, 352, 900, 368, 376, 672, 392, 600

Offset: 1

Omar E. Pol, Jan 15 2009

a(n) is even for all n.

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

Cf. A154720, A154721, A154723, A154783, A154785, A154786.

isnotcomp:=proc(n)return (n=1 or isprime(n)) end:
for n from 1 to 50 do rsum:=0: for k from 1 to 2*n-1 do if(not k=n and (isnotcomp(k) and isnotcomp(2*n-k)))then rsum:=rsum+k:fi:od: printf("%d, ",rsum):od: # Nathaniel Johnston, Apr 19 2011

a(n) = A154783(n) - n.

Edited by Omar E. Pol, Jan 17 2009

a(11)-a(50) from Nathaniel Johnston, Apr 19 2011

A154804 Number of ways to represent 2*n as the sum of two distinct primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 3, 3, 2, 4, 2, 2, 4, 3, 3, 4, 2, 3, 5, 4, 3, 6, 4, 3, 6, 3, 3, 7, 3, 5, 6, 3, 5, 7, 5, 5, 7, 5, 4, 9, 4, 4, 10, 4, 4, 7, 4, 6, 9, 6, 5, 9, 7, 7, 11, 6, 5, 12, 3, 5, 10, 4, 7, 10, 5, 5, 9, 8, 7, 11, 5, 5, 13, 5, 8, 11, 5, 8, 10, 6, 5, 14, 9, 6, 12, 7, 6, 15, 7, 8, 13, 5, 8, 12, 7, 9

Offset: 1

Omar E. Pol, Jan 16 2009

nonn

Number of ways to represent 2*n as the sum of two distinct noncomposite numbers. - Omar E. Pol, Dec 11 2024

Cf. A008578, A154720, A154721, A154722, A154723.

a(n) = A101264(n-1) + A061357(n). [From R. J. Mathar, Jan 21 2009]

a(n) = A001031(n) - A080339(n).

More terms from R. J. Mathar, Jan 21 2009

Edited by Franklin T. Adams-Watters, Jan 31 2009

A154799 Records in A154804.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 20, 21, 24, 28, 31, 33, 41, 42, 44, 52, 58, 69, 73, 76, 83, 92, 98, 115, 129, 138, 139, 154, 164, 165, 166, 172, 190, 198, 218, 223, 241, 268, 274, 292, 304, 330, 331, 341, 362, 394, 434, 447, 448, 466, 478

Offset: 1

Omar E. Pol, Feb 01 2009

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..207

Cf. A008578, A154804, A154720, A154721.

read("transforms") ; A008578 := proc(n) RETURN( n=1 or isprime(n) ) ; end : A154804 := proc(n) local a,d; a := 0 ; for d from 1 to n-1 do if A008578(n-d) and A008578(n+d) then a := a+1; fi; od: a ; end: a804 := [seq(A154804(n),n=1..5000)] ; r := RECORDS(a804)[1] ; # R. J. Mathar, Mar 14 2009

More terms from R. J. Mathar, Mar 14 2009

a(53) and beyond from Michael S. Branicky, Dec 11 2024

A154787 a(n) = A061357(n)*n = A154786(n)/2.

Original entry on oeis.org

0, 0, 0, 4, 5, 6, 7, 16, 18, 20, 22, 36, 26, 28, 45, 32, 51, 72, 19, 60, 84, 66, 69, 120, 100, 78, 135, 84, 87, 180, 62, 160, 198, 68, 175, 216, 148, 190, 273, 160, 164, 336, 172, 176, 405, 184, 188, 336, 147, 300, 408, 260, 265, 432, 330, 392, 570

Offset: 1

Omar E. Pol, Jan 20 2009

nonn

Cf. A061357, A154720, A154724, A154725, A154726, A154786.

A154791 Triangle read by rows in which row n lists all the pairs of noncomposite numbers that are equidistant from n, with 0's inserted, as shown below in the example.

Original entry on oeis.org

0, 1, 3, 1, 0, 5, 1, 3, 5, 7, 0, 3, 0, 7, 0, 1, 0, 5, 7, 0, 11, 1, 3, 0, 0, 0, 11, 13, 0, 3, 5, 0, 0, 11, 13, 0, 1, 0, 5, 7, 0, 11, 13, 0, 17, 1, 3, 0, 7, 0, 0, 13, 0, 17, 19, 0, 3, 5, 0, 0, 0, 0, 0, 17, 19, 0, 1, 0, 5, 7, 0, 11, 13, 0, 17, 19, 0, 23

Offset: 1

Omar E. Pol, Jan 25 2009

			Triangle begins:
n ....... Pairs
1 ..........0
2 ........ 1,3
3 ....... 1,0,5
4 ...... 1,3,5,7
5 ..... 0,3,0,7,0
6 .... 1,0,5,7,0,11

Cf. A008578, A154720, A154721, A154722, A154723.

iscomp(n) = (n!=1) && !isprime(n);
vecp(n) = {v = vector(n, i, 2*i-1); for (i = 1, n\2, if (iscomp(v[i]) || iscomp(v[n-i+1]), v[i] = 0; v[n-i+1] = 0);); if ((n % 2), v[n\2+1] = 0); v;}
trgp(nn) = {for (n = 1, nn, v = vecp(n); for (k = 1, n, print1(v[k], ", ");); print(););} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A154800 List of pairs of primes that are equidistant from and nearest to n, or the pair (0,0) if there is no such pair.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 5, 3, 7, 5, 7, 3, 11, 5, 11, 7, 11, 7, 13, 5, 17, 11, 13, 7, 19, 11, 17, 13, 17, 13, 19, 11, 23, 17, 19, 7, 31, 17, 23, 19, 23, 13, 31, 17, 29, 19, 29, 19, 31, 23, 29, 23, 31, 19, 37, 17, 41, 29, 31, 19, 43, 23, 41, 29, 37, 31, 37, 29, 41, 31, 41, 31, 43, 29, 47

Offset: 1

Omar E. Pol, Jan 15 2009, Jan 18 2009

Conjecture: There are only six 0's in this sequence.

			Array begins:
n ............... Pair
1 .............. 0 . 0
2 .............. 0 . 0
3 .............. 0 . 0
4 .............. 3 . 5
5 ............ 3 . . . 7
6 .............. 5 . 7
7 ........ 3 . . . . . . . 11
8 .......... 5 . . . . . 11
9 ............ 7 . . . 11
10 ......... 7 . . . . . 13

Cf. A154720, A154721, A154722, A154723, A154724, A154725, A154726, A154727.

A154800 := proc(n) local d; for d from 1 to n-1 do if isprime(n-d) and isprime(n+d) then printf("%d,%d,",n-d,n+d); RETURN() ; fi; od: printf("0,0,") ; end: seq(A154800(n),n=1..80) ; # R. J. Mathar, Jul 13 2009

More terms from R. J. Mathar, Jul 13 2009

A154891 Where records occurs in A154804.

Original entry on oeis.org

1, 2, 4, 9, 12, 21, 24, 30, 42, 45, 57, 60, 75, 84, 90, 105, 135, 150, 165, 195, 210, 255, 315, 330, 390, 420, 525, 630, 735, 825, 840, 945, 1050, 1155, 1365, 1575, 1680, 1785, 1995, 2100, 2145, 2205, 2310, 2625, 2730, 3045, 3255, 3465, 3990, 4095, 4515, 4620

Offset: 1

Omar E. Pol, Feb 01 2009

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..207

Cf. A008578, A154804, A154720, A154721, A154799.

A008578 := proc(n) RETURN( n=1 or isprime(n) ) ; end : A154804 := proc(n) local a,d; a := 0 ; for d from 1 to n-1 do if A008578(n-d) and A008578(n+d) then a := a+1; fi; od: a ; end: a804 := [seq(A154804(n),n=1..4000)] ; r := RECORDS(a804)[2] ; # R. J. Mathar, Mar 14 2009

More terms from R. J. Mathar, Mar 14 2009

a(50) and beyond from Michael S. Branicky, Dec 11 2024

A154792 Triangle read by rows in which row n lists the pair of noncomposite numbers that are equidistant from and nearest to n, with 0's inserted, as shown below in the example.

Original entry on oeis.org

0, 1, 3, 1, 0, 5, 0, 3, 5, 0, 0, 3, 0, 7, 0, 0, 0, 5, 7, 0, 0, 0, 3, 0, 0, 0, 11, 0, 0, 0, 5, 0, 0, 11, 0, 0, 0, 0, 0, 7, 0, 11, 0, 0, 0, 0, 0, 0, 7, 0, 0, 13, 0, 0, 0

Offset: 1

Omar E. Pol, Jan 25 2009

			Triangle begins:
n ........ Pair
1 ..........0
2 ........ 1,3
3 ....... 1,0,5
4 ...... 0,3,5,0
5 ..... 0,3,0,7,0
6 .... 0,0,5,7,0,0

Cf. A008578, A154720, A154721, A154722, A154723, A154791.

cp's OEIS Frontend

A184995 Irregular triangle T, read by rows, in which row n lists the primes p <= n such that 2n-p is also prime.

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A154784 Row sums of triangle in A154721.

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A154804 Number of ways to represent 2*n as the sum of two distinct primes (counting 1 as a prime).

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A154799 Records in A154804.

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A154787 a(n) = A061357(n)*n = A154786(n)/2.

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A154791 Triangle read by rows in which row n lists all the pairs of noncomposite numbers that are equidistant from n, with 0's inserted, as shown below in the example.

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A154800 List of pairs of primes that are equidistant from and nearest to n, or the pair (0,0) if there is no such pair.

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Maple

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A154891 Where records occurs in A154804.

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A154792 Triangle read by rows in which row n lists the pair of noncomposite numbers that are equidistant from and nearest to n, with 0's inserted, as shown below in the example.

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