A283814 Irregular triangle read by rows in which n-th row lists the numbers m such that 2*prime(m) can be represented as the sum of two primes in exactly n ways.

1, 2, 3, 4, 8, 5, 6, 11, 7, 9, 10, 18, 12, 13, 14, 15, 22, 16, 17, 19, 21, 23, 24, 25, 27, 29, 30, 34, 38, 46, 20, 28, 42, 26, 31, 32, 36, 37, 40, 50, 41, 43, 58, 33, 35, 39, 45, 47, 52, 53, 59, 44, 48, 49, 65, 51, 61, 62, 55, 57, 60, 66, 67, 70, 85, 54, 56, 63, 68, 72, 73, 75, 77, 79, 64, 76, 78, 80, 81, 83

Offset: 1

Author

Zak Seidov, Mar 17 2017

Keywords

• nonn
• tabf

Comments

From b116619.txt it seems that the sequence is correct at least for first 677 terms (first 100 rows of triangle). But as it is usual in number theory better consider this sequence as conjectured.

Lengths of first 100 rows of triangle (see a283814.txt): {2,3,3,4,5,5,8,3,7,3,8,4,3,7,8,1,10,7,6,9,3,7,6,3,4,7,13,4,6,7,7,9,7,8,8,3,8,8,5,5,5,11,5,10,3,6,8,10,5,8,5,9,6,9,6,7,10,6,6,6,8,5,7,12,11,6,8,6,9,4,12,6,8,5,5,5,11,10,13,7,7,10,9,7,4,9,7,5,4,8,7,6,10,7,6,10,6,10,6,6}.

Examples

			3rd row is {5,6,11} because only the 5th, 6th and 11th primes can be represented as the sum of 2 primes in exactly 3 ways:
n=3: 2*prime(5) = 2*11 = 22 = 3 + 19 = 5 + 17 = 11 + 11,
2*prime(6) = 2*13 = 26 = 3 + 23 = 7 + 19 = 13 + 13,
2*prime(11) = 2*31 = 62 = 3 + 59 = 7 + 19 = 19 + 43 = 31 + 31.
		

Links

• Zak Seidov, First 100 rows of the triangle.

Crossrefs

Cf. A116619 (number of ways of representing 2*prime(n) as the sum of two primes).

Programs

Mathematica

A116619=Table[Count[PrimeQ[2*Prime[n]-Prime[Range[n]]],True],{n,1000}];
Flatten[Position[A116619,#]& /@ Range[100]]