A337739 Terms of A083209 with a record number of divisors.

6, 12, 56, 176, 550, 2752, 3230, 8925, 351351

Offset: 1

Author

Amiram Eldar, Sep 17 2020

Keywords

• nonn
• more

Comments

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.

The corresponding numbers of divisors are 4, 6, 8, 10, 12, 14, 16, 24, 48, ...

a(10) > 1.8*10^6.

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

Examples

			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.
		

Crossrefs

Cf. A000005, A000203, A023196, A083207, A083209, A335008, A337738.

Programs

Mathematica

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