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%I #14 Aug 26 2024 18:51:01
%S 6,28,55,496,666,780,1540,2145,6441,6903,8128,15051,21736,36585,44551,
%T 232903,234955,644680,2258875,3186550,3462396,6211050,22174470,
%U 33550336,48437403,62591266,107538445,134898525,153554050,624157446,1309312378,1339937028
%N Triangular numbers whose sum of divisors is an oblong number.
%C Includes all the even perfect numbers.
%C The indices of these triangular numbers are 3, 7, 10, 31, 36, 39, 55, 65, 113, 117, 127, 173, 208, 270, 298, 682, 685, 1135, 2125, 2524, 2631, 3524, 6659, 8191, 9842, 11188, 14665, 16425, 17524, 35331, 51172, 51767, 52019, 52486, 58993, 65585, 97532.
%C The indices of the corresponding oblong numbers are 3, 7, 8, 31, 38, 48, 63, 63, 95, 104, 127, 144, 224, 255, 224, 512, 575, 1215, 1728, 2448, 3072, 3968, 7695, 8191, 9215, 9792, 12159, 15872, 17576, 37296, 46656, 58239, 63855, 40959, 46080, 62720, 102960.
%C Number of terms < 10^k, k=1,2,3...: 1, 3, 6, 11, 15, 18, 22, 26, 30, 40, 52, 64, 80, 90, 110, 128, ..., . - _Robert G. Wilson v_, Jul 31 2018
%H Robert G. Wilson v, <a href="/A317478/b317478.txt">Table of n, a(n) for n = 1..129</a>
%e 55 is in the sequence since sigma(55) = 72 = 8 * 9 is an oblong number.
%t tri[n_] := n(n+1)/2; aQ[n_] := IntegerQ[Sqrt[4 * DivisorSigma[1, tri[n]] + 1]]; tri[Select[Range[52000], aQ]]
%t Module[{nn=60000,obno},obno=Table[n(n+1),{n,nn}];Select[Accumulate[Range[nn]],MemberQ[ obno,DivisorSigma[1,#]]&]] (* _Harvey P. Dale_, Aug 26 2024 *)
%Y Cf. A000203, A000217, A000396, A002378, A083674, A083675, A175849.
%Y Intersection of A000217 and A236387. - _Michel Marcus_, Jul 30 2018
%K nonn,easy
%O 1,1
%A _Amiram Eldar_, Jul 29 2018