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The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of distinct prime factors.
1

%I #14 Aug 25 2024 02:49:20

%S 1,6,6,6,6,6,6,6,103327500,1977230170,214596675756,2796022463625,

%T 32291440570378,32291440570378,549091407981061,4680505857819681,

%U 96914999755056255,96914999755056255

%N The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of distinct prime factors.

%e a(8) = 6 because 6 is the smallest triangular number that begins a run of 8 consecutive triangular numbers (6, 10, 15, 21, 28, 36, 45, 55) with the same number of distinct prime factors, i.e. 2.

%e a(9) = 103327500 because 103327500 is the smallest triangular number that begins a run of 9 consecutive triangular numbers (103327500, 103341876, 103356253, 103370631, 103385010, 103399390, 103413771, 103428153, 103442536) with the same number of distinct prime factors, i.e. 5.

%Y Cf. A000217, A069903, A076551.

%Y Cf. A375657.

%K nonn,hard,more

%O 1,2

%A _Shyam Sunder Gupta_, Aug 23 2024