A277729 Irregular triangle read by rows: T(n,k) = number of times a gap of k occurs between the first n successive primes.

1, 1, 1, 1, 2, 1, 2, 0, 1, 1, 3, 0, 1, 1, 3, 0, 2, 1, 4, 0, 2, 1, 4, 0, 3, 1, 4, 0, 3, 0, 1, 1, 5, 0, 3, 0, 1, 1, 5, 0, 3, 0, 2, 1, 5, 0, 4, 0, 2, 1, 6, 0, 4, 0, 2, 1, 6, 0, 5, 0, 2, 1, 6, 0, 5, 0, 3, 1, 6, 0, 5, 0, 4, 1, 7, 0, 5, 0, 4, 1, 7, 0, 5, 0, 5, 1, 7, 0, 6, 0, 5, 1, 8, 0, 6, 0, 5, 1, 8, 0, 6, 0, 6, 1, 8, 0, 7, 0, 6, 1, 8, 0, 7, 0, 7, 1, 8, 0, 7, 0, 7, 0, 1, 1, 8, 0, 8, 0, 7, 0, 1, 1, 9, 0

Offset: 2

Author

N. J. A. Sloane, Nov 06 2016

Keywords

• nonn
• tabf

Examples

			Triangle begins:
1,
1, 1,
1, 2,
1, 2, 0, 1,
1, 3, 0, 1, <- gaps in 2,3,5,7,11,13 are 1, 2 (3 times), 4 (once)
1, 3, 0, 2,
1, 4, 0, 2,
1, 4, 0, 3,
1, 4, 0, 3, 0, 1,
1, 5, 0, 3, 0, 1,
1, 5, 0, 3, 0, 2,
1, 5, 0, 4, 0, 2,
1, 6, 0, 4, 0, 2,
1, 6, 0, 5, 0, 2,
1, 6, 0, 5, 0, 3,
1, 6, 0, 5, 0, 4,
1, 7, 0, 5, 0, 4,
1, 7, 0, 5, 0, 5,
1, 7, 0, 6, 0, 5,
1, 8, 0, 6, 0, 5,
...
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10032 (rows 2 to 410, flattened)

Crossrefs

Cf. A277730, A213930.

Programs

Maple

N:= 30: # for rows 2 to N
res:= 1; p:= 3; R:= <1>;
for n from 2 to N do
  pp:= nextprime(p);
  d:= pp - p;
  p:= pp;
  if d <= LinearAlgebra:-Dimension(R) then
    R[d]:= R[d]+1
  else
    R(d):= 1
  fi;
  res:= res, op(convert(R,list));
od:
res;  # Robert Israel, Nov 16 2016