1,1

1: q=p+2n is also a prime, although not necessarily the next after p;

2: the k-th position of the n-th row gives is a prime p such that the number of further primes between p and q=p+2n (not counting p and q) is k-1;

3: the primes p are the smallest with these properties.

Thus each row onlyq contains primes. The first term in the n-th row is A000230(n). The last one in the same row is A020483(n). The length of the n-th row is Pi[2n+A020483(n)]-Pi[A020483(n)].

The table begins as follows:

3,

7,3,

23,5,

89,23,3,

139,19,7,3,

199,47,17,5,

113,83,23,17,3,

For example, suppose n = 50: d=2n=100; the 50th row consists of 24 terms as follows:

{396733, 58789, 142993, 38461, 37699, 7351, 5881, 1327, 2557, 1879, 1621, 1117, 463, 457, 283, 331, 211, 127, 73, 67, 31, 0, 7, 3};

A000230[50]=396733, A020483[50]=3; between 143093 and 142993 two primes {143053,143063} occur because 142993 is the (2+1)rd entry in the 50th row.

The length of n-th row is Pi[100+3]-Pi[3]-1=Pi[106]-Pi[3]=27-2-1=24, number of primes between 103 and 3 is 23 (not counting 103 and 3).

Program to generate the 19-th row: cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=38, k=0, mxc=Ceiling[d/3]; vg=PrimePi[30593]} t=Table[0, {mxc}]; t1=Table[0, {mxc}]; Do[s=cp[1+Prime[n], Prime[n]+d-1]; np=d+Prime[n]; If[PrimeQ[np]&&s<(1+mxc)&&t[[s+1]]==0, t[[s+1]]=n; t1[[s+1]]=Prime[n]], {n, 1, 5000}]; {t, t1}

Cf. A000720, A000230, A020483, A086155, A086138-A086149, A086155.

Sequence in context: A096385 A088837 A019158 this_sequence A049479 A125314 A050393

Adjacent sequences: A086150 A086151 A086152 this_sequence A086154 A086155 A086156

nonn,tabf

Labos E. (labos(AT)ana.sote.hu), Aug 08 2003