A335198 Infinitary Zumkeller numbers (A335197) whose number of divisors is not a power of 2.

60, 72, 90, 96, 150, 294, 360, 420, 480, 486, 504, 540, 600, 630, 660, 672, 726, 756, 780, 792, 864, 924, 936, 960, 990, 1014, 1020, 1050, 1056, 1092, 1120, 1140, 1152, 1170, 1176, 1188, 1224, 1248, 1344, 1350, 1368, 1380, 1386, 1400, 1428, 1440, 1470, 1500, 1530

Offset: 1

Author

Amiram Eldar, May 26 2020

Keywords

• nonn

Comments

Zumkeller numbers (A083207) whose number of divisors is a power of 2 (A036537) are also infinitary Zumkeller numbers (A335197), since all of their divisors are infinitary.

Examples

			72 is a term since its set of infinitary divisors, {1, 2, 4, 8, 9, 18, 36, 72}, can be partitioned into the two disjoint sets, {1, 2, 72} and {4, 8, 9, 18, 36}, whose sum is equal: 1 + 2 + 72 = 4 + 8 + 9 + 18 + 36 = 75.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A335197.

Cf. A083207, A036537.

Programs

Mathematica

infdivs[n_] := If[n == 1, {1}, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infZumQ[n_] := Module[{d = infdivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[1500], ! pow2Q[DivisorSigma[0, #]] && infZumQ[#] &] (* after Michael De Vlieger at A077609 *)