cp's OEIS Frontend

A333756 a(n) is the start of the first run of exactly n consecutive nonsquarefree triangular numbers.

%I A333756 #16 Apr 08 2020 07:33:23
%S A333756 496,28,120,276,4560,28680,116886,1460295903,1423828,16672425,
%T A333756 40046775,969738780,5300947095,29604866115,70439870130,4074768806430,
%U A333756 8073317216328,2299554739121745,7099676667360280,71866989786336690,9087907667048616,337295518424356416
%N A333756 a(n) is the start of the first run of exactly n consecutive nonsquarefree triangular numbers.
%C A333756 For every positive integer k, the k-th triangular number T(k) = A000217(k) = k*(k+1)/2 can be written as the product of two comprime factors, f1 and f2, where f2 = 2*f1 +- 1; e.g.,
%C A333756     k  T(k)    f1  f2
%C A333756     -  ----    --  --
%C A333756     1     1  =  1 * 1
%C A333756     2     3  =  1 * 3
%C A333756     3     6  =  2 * 3
%C A333756     4    10  =  2 * 5
%C A333756 Since f1 and f2 are coprime, T(k) is squarefree iff both f1 and f2 are squarefree. Every pair of consecutive triangular numbers shares either the factor f1 or f2, so if T(k) is nonsquarefree, then at least one of T(k-1) and T(k+1) must also be nonsquarefree. Thus, no "run" of exactly one nonsquarefree triangular number exists (so a(1) does not exist).
%H A333756 Giovanni Resta, <a href="/A333756/b333756.txt">Table of n, a(n) for n = 2..29</a> (terms < 5*10^25)
%e A333756 The 30th through 33rd triangular numbers are
%e A333756   T(30) = 465 = 3 * 5 * 13   (squarefree),
%e A333756   T(31) = 496 = 2^4 * 31     (nonsquarefree),
%e A333756   T(32) = 528 = 2^4 * 3 * 11 (nonsquarefree), and
%e A333756   T(33) = 561 = 3 * 11 * 17  (squarefree),
%e A333756 so 496 begins a run of exactly two consecutive nonsquarefree triangular numbers. Since 496 is the smallest such triangular number, a(2) = 496.
%Y A333756 Cf. A000217 (triangular numbers), A061304 (squarefree triangular numbers), A061900 (triangular numbers that are not squarefree).
%K A333756 nonn
%O A333756 2,1
%A A333756 _Jon E. Schoenfield_, Apr 04 2020
%E A333756 Terms a(23) and beyond from _Giovanni Resta_, Apr 07 2020