A306196 Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.

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

Offset: 2

Author

Juri-Stepan Gerasimov, Jan 28 2019

Keywords

• nonn
• easy
• tabf

Comments

Conjectures:

(i) 1 <= A035026(n) <= (n-th row length of this triangle) for n >= 2;

(ii) a(n,1) < A171637(n,1) for n >= 4.

Numbers m such that m-th row length of this triangle is equal to A000720(m): 1, 2, 11, 13, 25, 56, 60, ...

Examples

			Row 2 = [2] because 2*2 = 2 + 2;
Row 3 = [3] because 2*3 = 3 + 3;
Row 4 = [2,3,5] because 2*4 - 2 = 6 = 2*3 and 2*4 = 3 + 5;
Row 5 = [3,5,7] because 2*5 = 3 + 7 = 5 + 5.
The table starts:
  2;
  3;
  2,  3,  5;
  3,  5,  7;
  2,  5,  7;
  2,  3,  5,  7, 11;
  3,  5,  7, 11, 13;
  3,  5,  7, 11, 13;
  2,  3,  5,  7, 11, 13, 17;
  2,  3,  5,  7, 11, 13, 17, 19;
  2,  3,  5,  7, 11, 13, 17, 19;
  2,  3,  5,  7, 11, 13, 17, 19, 23;
  3,  5, 11, 13, 17, 23;
  2,  7, 11, 13, 17, 19, 23;
  2,  3,  5, 11, 13, 17, 19, 23, 29;
		

Crossrefs

Supersequence of A171637.

Cf. A000720, A020481, A035026, A306247, A306261.

Programs

PARI

isok(k,n) = {if (isprime(2*n-k), pf = factor(k)[,1]; for (j=1, #pf, if (isprime(2*n-pf[j]), return (1));););}
row(n) = {my(v = []); for (k=1, 2*n, if (isok(k,n), v = concat(v, 2*n-k))); vecsort(v);} \\ Michel Marcus, Mar 02 2019