A349216 - OEIS

%I #27 Feb 16 2022 01:56:04

%S 0,0,1,2,4,8,13,20,30,40,53,70,88,110,137,166,200,240,281,328,382,438,

%T 501,572,646,728,819,910,1010,1120,1233,1356,1490,1628,1777,1938,2100,

%U 2274,2461,2652,2856,3074,3297,3534,3786,4040,4309,4594,4884,5190,5513,5842,6188,6552,6917

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

%C 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.

%C 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).

%H Coen del Valle and Peter J. Dukes, <a href="https://arxiv.org/abs/2201.00897">Balancing permuted copies of multigraphs and integer matrices</a>, arXiv:2201.00897 [math.CO], 2022.

%e 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).

%t 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 *)

%o (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

%o (SageMath)

%o def a(n):

%o     t=3^ceil(log(n,3))

%o     counter=0

%o     for w in range(n):

%o         for v in range(w):

%o             for u in range(v):

%o                 if min(gcd(w-u,3^t),gcd(w-v,3^t))<gcd(2*w-u-v,3^t):

%o                     counter+=1

%o     return counter

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

%K nonn,base

%O 1,4

%A _Peter J. Dukes_, Nov 10 2021