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A385202 Irregular triangle read by rows: let S be an ordered set of nondivisors of n such that a and b belong to S if a + b = n. T(n,k) is the k-th member of S. If S is empty, T(n,k) = 0.

%I A385202 #6 Jun 26 2025 23:00:02
%S A385202 0,0,0,0,2,3,0,2,3,4,5,3,5,2,4,5,7,3,4,6,7,2,3,4,5,6,7,8,9,5,7,2,3,4,
%T A385202 5,6,7,8,9,10,11,3,4,5,6,8,9,10,11,2,4,6,7,8,9,11,13,3,5,6,7,9,10,11,
%U A385202 13,2,3,4,5,6,7,8,9,10,11,12,13,14,15,4,5,7,8,10,11,13,14
%N A385202 Irregular triangle read by rows: let S be an ordered set of nondivisors of n such that a and b belong to S if a + b = n. T(n,k) is the k-th member of S. If S is empty, T(n,k) = 0.
%F A385202 n = T(n, m) + T(n, k-(m-1)), 1 <= m <= k, for every row of length k.
%F A385202 S defined as in the name, n - |S| = A086369(n).
%e A385202  n | Triangle begins:
%e A385202 ---+-----------------
%e A385202  1 | {}
%e A385202  2 | {}
%e A385202  3 | {}
%e A385202  4 | {}
%e A385202  5 | {2, 3}
%e A385202  6 | {}
%e A385202  7 | {2, 3, 4, 5}
%e A385202  8 | {3, 5}
%e A385202  9 | {2, 4, 5, 7}
%e A385202 10 | {3, 4, 6, 7}
%e A385202 11 | {2, 3, 4, 5, 6, 7, 8, 9}
%e A385202 12 | {5, 7}
%e A385202 13 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
%e A385202 14 | {3, 4, 5, 6, 8, 9, 10, 11}
%e A385202 15 | {2, 4, 6, 7, 8, 9, 11, 13}
%e A385202 16 | {3, 5, 6, 7, 9, 10, 11, 13}
%e A385202 17 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
%e A385202 18 | {4, 5, 7, 8, 10, 11, 13, 14}
%e A385202 19 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}
%e A385202 20 | {3, 6, 7, 8, 9, 11, 12, 13, 14, 17}
%e A385202 21 | {2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 19}
%e A385202 22 | {3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19}
%e A385202 23 | {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
%e A385202 24 | {5, 7, 9, 10, 11, 13, 14, 15, 17, 19}
%e A385202 25 | {2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23}
%o A385202 (PARI) T(n,k) = my(S); S = select(x -> setsearch(divisors(n), x)==0 && setsearch(divisors(n), n-x)==0, [1..n]); if(k <= #S, S[k], 0) \\ function made to output 0 if k exceeds the size of S to avoid breaking
%Y A385202 Cf. A086369.
%K A385202 nonn,tabf
%O A385202 1,5
%A A385202 _Miles Englezou_, Jun 21 2025