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Numbers that start a run of four consecutive triangular numbers with four distinct prime factors.
1

%I #18 Dec 23 2021 12:38:19

%S 16653,16836,17020,17205,17391,27495,29890,30135,50721,51040,51360,

%T 51681,70125,81003,81406,84255,84666,85078,85491,85905,89676,90100,

%U 110215,110685,111156,142311,181503,214185,222111,222778,305371,306153,344865,345696,355746,356590

%N Numbers that start a run of four consecutive triangular numbers with four distinct prime factors.

%e a(1) = 16653 because 16653 is the smallest number in the first set of four consecutive triangular numbers with four distinct prime factors, i.e., 16653 = 3*7*13*61, 16836 = 2^2*3*23*61, 17020 = 2^2*5*23*37, 17205 = 3*5*31*37.

%t t[n_] := n*(n + 1)/2; q[n_] := PrimeNu[n] == 4; Select[Partition[t /@ Range[1000], 4, 1], AllTrue[#, q] &][[;; , 1]] (* _Amiram Eldar_, Nov 29 2021 *)

%Y Cf. A000217, A128905, A349539.

%K nonn

%O 1,1

%A _Shyam Sunder Gupta_, Nov 29 2021