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A277730 Irregular triangle read by rows: T(n,k) = number of times a gap of 2k occurs between the first n successive odd primes.

%I A277730 #22 Dec 26 2024 10:17:06
%S A277730 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 A277730 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 A277730 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 A277730 Irregular triangle read by rows: T(n,k) = number of times a gap of 2k occurs between the first n successive odd primes.
%C A277730 T(m, 2) = max{T(m, k): k >= 1} for m >= 100. - _Ya-Ping Lu_, Dec 25 2024
%H A277730 Lars Blomberg, <a href="/A277730/b277730.txt">Table of n, a(n) for n = 2..9997</a> (the first 703 rows)
%e A277730 Triangle begins:
%e A277730   1,
%e A277730   2,
%e A277730   2, 1,
%e A277730   3, 1, <- gaps in 3,5,7,11,13 are 2 (3 times), 4 (once)
%e A277730   3, 2,
%e A277730   4, 2,
%e A277730   4, 3,
%e A277730   4, 3, 1,
%e A277730   5, 3, 1,
%e A277730   5, 3, 2,
%e A277730   5, 4, 2,
%e A277730   6, 4, 2,
%e A277730   6, 5, 2,
%e A277730   6, 5, 3,
%e A277730   6, 5, 4,
%e A277730   7, 5, 4,
%e A277730   7, 5, 5,
%e A277730   7, 6, 5,
%e A277730   8, 6, 5,
%e A277730   ...
%o A277730 (Python)
%o A277730 from sympy import nextprime; p = 3; L = []
%o A277730 for n in range(2, 32):
%o A277730     np = nextprime(p); k = (np - p)//2
%o A277730     if len(L) < k: {L.append(0) for i in range(len(L), k-1)}; L.append(1)
%o A277730     else: L[k-1] += 1
%o A277730 print(*L, sep =", ", end = ", "); p = np  # _Ya-Ping Lu_, Dec 25 2024
%Y A277730 Cf. A277729, A213930.
%K A277730 nonn,look,tabf
%O A277730 2,2
%A A277730 _N. J. A. Sloane_, Nov 06 2016