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%I #18 Jun 15 2021 10:37:06
%S 6,12,56,176,550,2752,3230,8925,351351
%N Terms of A083209 with a record number of divisors.
%C Zumkeller numbers (A083207) which can be partitioned into two disjoint sets with an equal sum in a single way, and having a record number of divisors.
%C The corresponding numbers of divisors are 4, 6, 8, 10, 12, 14, 16, 24, 48, ...
%C a(10) > 1.8*10^6.
%C Per a comment by _T. D. Noe_ in A083209 we have a(10) <= 2^24 * 11184829 = 187650292056064 and this sequence is infinite. - _David A. Corneth_, May 19 2021
%e The first 5 terms of A083209 are 6, 12, 20, 28, 56. Their numbers of divisors are 4, 6, 6, 6, 8. The record values, 4, 6 and 8 occur at 6, 12 and 56.
%t zumsingleQ[n_] := Module[{d = Divisors[n], sum, x}, sum = Plus @@ d; sum >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]; dm = 0; s = {}; Do[d = DivisorSigma[0, n]; If[d > dm, q = zumsingleQ[n]; If[q && d > dm, dm = d; AppendTo[s, n]]], {n, 1, 10^4}]; s
%Y Cf. A000005, A000203, A023196, A083207, A083209, A335008, A337738.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Sep 17 2020
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