A366344 - OEIS

%I #10 Oct 12 2023 14:10:03

%S 1,1,2,1,1,2,3,1,4,1,3,2,1,3,2,3,1,4,5,1,6,1,5,2,9,1,8,1,9,2,5,3,10,3,

%T 3,4,1,5,2,5,1,6,3,5,4,3,3,10,1,9,2,9,1,8,7,1,10,1,7,2,15,1,12,1,15,2,

%U 7,3,14,3,5,4,25,1,18,1,25,2,27,1,16,1,27,2

%N Irregular triangle T(n, k), n >= 0, k = 1 or 2, read by rows; the n-th row contains two coprime positive integers whose prime factorizations are encoded in the ternary expansion of n (see Comments section for precise definition).

%C The encoding used here is related to that used for the Doudna sequence (A005940):

%C - for any pair (u, v) of coprime positive integers, the ternary expansion of the unique n >= 0 such that T(n, 1) = u and T(n, 2) = v is built as follows (from right to left):

%C    - for m = 1, 2, ..., let p be the m-th prime number,

%C      - if p neither divides u nor v then we add a 0,

%C      - if p divides u with multiplicity e then we add a run of e 1's,

%C      - if p divides v with multiplicity e then we add a run of e 2's,

%C      - we also insert an extra 0 between pairs of runs of 1's not separated by 2's and between pairs of runs of 2's not separated by 1's.

%C This encoding can be applied to any fixed base b >= 2 and will yield a bijection from the nonnegative integers to the set of tuples of b-1 pairwise coprime positive integers.

%C The case b = 2 corresponds (up to the offset) to the Doudna sequence (A005940).

%C The sequence n -> T(n, 1) / T(n, 2) runs through all the reduced positive rationals exactly once.

%F T(n, 1) = 1 iff n belongs to A005823.

%F T(n, 2) = 1 iff n belongs to A005836.

%F T(A005836(n), 1) = A005940(n+1).

%F T(A005823(n), 2) = A005940(n+1).

%F A001222(T(n, 1)) = A062756(n).

%F A001222(T(n, 2)) = A081603(n).

%F A001222(T(n, 1) * T(n, 2)) = A160384(n).

%F T(A004488(n), 1) = T(n, 2).

%F T(A004488(n), 2) = T(n, 1).

%F T((3^e - 1)/2, 1) = 2^e for any e >= 0.

%F T(3^e - 1, 2) = 2^e for any e >= 0.

%F T(3^e, 1) = A000040(e + 1) for any e >= 0.

%F T(2 * 3^e, 2) = A000040(e + 1) for any e >= 0.

%F T(3*n, k) = A003961(T(n, k)).

%e Triangle T(n, k) begins (alongside the ternary expansion of n):

%e   n   n-th row  ter(n)

%e   --  --------  ------

%e    0  [1, 1]         0

%e    1  [2, 1]         1

%e    2  [1, 2]         2

%e    3  [3, 1]        10

%e    4  [4, 1]        11

%e    5  [3, 2]        12

%e    6  [1, 3]        20

%e    7  [2, 3]        21

%e    8  [1, 4]        22

%e    9  [5, 1]       100

%e   10  [6, 1]       101

%e   11  [5, 2]       102

%e   12  [9, 1]       110

%e   13  [8, 1]       111

%e   14  [9, 2]       112

%e   15  [5, 3]       120

%e   16  [10, 3]      121

%e   17  [3, 4]       122

%o (PARI) row(n, b = 3) = { my (r = vector(b-1, d, 1), g = 0, t = 0); while (n, my (d = n % b); n \= b; g++; if (d, my (e = 1); while (n % b == d, e++; n \= b;); if (t==d, g--, t = d); r[d] *= prime(g)^e;);); return (r); }

%Y Cf. A000040, A001222, A003961, A004488, A005823, A005836, A005940, A062756, A081603, A160384.

%K nonn,base,tabf

%O 0,3

%A _Rémy Sigrist_, Oct 07 2023