A349216 Number of ternary triples (u,v,w) with 1 <= u < v < w <= n.

0, 0, 1, 2, 4, 8, 13, 20, 30, 40, 53, 70, 88, 110, 137, 166, 200, 240, 281, 328, 382, 438, 501, 572, 646, 728, 819, 910, 1010, 1120, 1233, 1356, 1490, 1628, 1777, 1938, 2100, 2274, 2461, 2652, 2856, 3074, 3297, 3534, 3786, 4040, 4309, 4594, 4884, 5190, 5513, 5842, 6188, 6552, 6917

Offset: 1

Author

Peter J. Dukes, Nov 10 2021

Keywords

• nonn
• base

Comments

A triple of integers (u,v,w) is a ternary triple if in the ternary expansions of u,v,w, all three disagree at the least significant position at which any two disagree.

Equivalently, (u,v,w) is a ternary triple if the highest power of three dividing 2w-u-v is greater than the highest power of three dividing gcd(w-u,w-v).

Examples

			For n = 7 the 13 ternary triples are (1, 2, 3), (2, 3, 4), (1, 3, 5), (3, 4, 5), (1, 2, 6), (2, 4, 6), (1, 5, 6), (4, 5, 6), (2, 3, 7), (1, 4, 7), (3, 5, 7), (2, 6, 7), (5, 6, 7).
		

Links

• Coen del Valle and Peter J. Dukes, Balancing permuted copies of multigraphs and integer matrices, arXiv:2201.00897 [math.CO], 2022.

Crossrefs

Cf. A061866, A005704, A038500, A007089, A007997.

Programs

Mathematica

Array[Sum[Sum[Sum[Boole[IntegerExponent[w + w - u - v, 3] > IntegerExponent[GCD[w - u, w - v], 3]], {u, (v - 1)}], {v, 2, (w - 1)}], {w, 3, #}] &, 55] (* Michael De Vlieger, Feb 15 2022 *)

PARI

A349216(n) = sum(w=3,n,sum(v=2,(w-1),sum(u=1,(v-1),valuation(w+w-u-v,3) > valuation(gcd(w-u,w-v),3)))); \\ Antti Karttunen, Nov 13 2021

SageMath

def a(n):
    t=3^ceil(log(n,3))
    counter=0
    for w in range(n):
        for v in range(w):
            for u in range(v):
                if min(gcd(w-u,3^t),gcd(w-v,3^t))