A277730 - OEIS

%I #22 Dec 26 2024 10:17:06

%S 1,2,2,1,3,1,3,2,4,2,4,3,4,3,1,5,3,1,5,3,2,5,4,2,6,4,2,6,5,2,6,5,3,6,

%T 5,4,7,5,4,7,5,5,7,6,5,8,6,5,8,6,6,8,7,6,8,7,7,8,7,7,1,8,8,7,1,9,8,7,

%U 1,9,9,7,1,10,9,7,1,10,10,7,1,10,10,7,1,0,0,1,10,11,7,1,0,0,1

%N Irregular triangle read by rows: T(n,k) = number of times a gap of 2k occurs between the first n successive odd primes.

%C T(m, 2) = max{T(m, k): k >= 1} for m >= 100. - _Ya-Ping Lu_, Dec 25 2024

%H Lars Blomberg, <a href="/A277730/b277730.txt">Table of n, a(n) for n = 2..9997</a> (the first 703 rows)

%e Triangle begins:

%e   1,

%e   2,

%e   2, 1,

%e   3, 1, <- gaps in 3,5,7,11,13 are 2 (3 times), 4 (once)

%e   3, 2,

%e   4, 2,

%e   4, 3,

%e   4, 3, 1,

%e   5, 3, 1,

%e   5, 3, 2,

%e   5, 4, 2,

%e   6, 4, 2,

%e   6, 5, 2,

%e   6, 5, 3,

%e   6, 5, 4,

%e   7, 5, 4,

%e   7, 5, 5,

%e   7, 6, 5,

%e   8, 6, 5,

%e   ...

%o (Python)

%o from sympy import nextprime; p = 3; L = []

%o for n in range(2, 32):

%o     np = nextprime(p); k = (np - p)//2

%o     if len(L) < k: {L.append(0) for i in range(len(L), k-1)}; L.append(1)

%o     else: L[k-1] += 1

%o print(*L, sep =", ", end = ", "); p = np  # _Ya-Ping Lu_, Dec 25 2024

%Y Cf. A277729, A213930.

%K nonn,look,tabf

%O 2,2

%A _N. J. A. Sloane_, Nov 06 2016