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

1,1

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.

Links

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

Example

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.

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 *)

Crossrefs

Subsequence of A335197.

Cf. A083207, A036537.

Sequence in context: A030630 A068350 A265712 * A335935 A347935 A347938

Adjacent sequences: A335195 A335196 A335197 * A335199 A335200 A335201

Keyword

nonn

Author

Amiram Eldar, May 26 2020

Status

approved