A293453 Zumkeller numbers k such that sigma(k)/2 is a Zumkeller number.

6, 24, 28, 42, 54, 56, 60, 78, 84, 88, 96, 102, 108, 114, 120, 126, 132, 140, 150, 168, 174, 176, 186, 198, 204, 216, 220, 222, 224, 228, 240, 246, 252, 258, 260, 264, 270, 276, 280, 294, 308, 312, 330, 336, 340, 342, 348, 350, 352, 354, 366, 368, 372, 378, 380, 384, 390, 396, 402

Offset

1,1

Comments

Subsequence of A083207 (Zumkeller numbers).

Conjecture: Any four consecutive Zumkeller numbers include at least one term of the present sequence (verified for the first 10^5 Zumkeller numbers).

The perfect numbers (A000396) are all trivially in this sequence.

Links

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

Example

The fourth Zumkeller number is 24, since sigma(24) = A000203(24) = 60, 60/2 = 30, and the divisors of 24 can be partitioned as 1 + 2 + 3 + 4 + 8 + 12 = 6 + 24 = 30.
In turn, 30 is also a Zumkeller number, as sigma(30)/2 = 72/2 = 36 and 1 + 2 + 3 + 5 + 10 + 15 = 6 + 30 = 36.
Therefore 24 is in this sequence.
But since 36 is not a Zumkeller number at all, 30 is not in this sequence.

Mathematica

zumkellerQ[n_] := Module[{d = Divisors[n], t, ds, x}, ds = Plus@@d; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; zn = Select[Range[1000], zumkellerQ] (* a code by T. D. Noe at A083207 *); Select[zn, zumkellerQ[DivisorSigma[1, #]/2] &]

Crossrefs

Cf. A000203, A000396, A083207.

Sequence in context: A349686 A069235 A175200 * A344754 A364977 A336641

Adjacent sequences: A293450 A293451 A293452 * A293454 A293455 A293456

Keyword

easy,nonn

Author

Ivan N. Ianakiev, Oct 09 2017

Status

approved