

A293453


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


1



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
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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

Table of n, a(n) for n=1..59.


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: A273124 A069235 A175200 * A118372 A263928 A219362
Adjacent sequences: A293450 A293451 A293452 * A293454 A293455 A293456


KEYWORD

easy,nonn


AUTHOR

Ivan N. Ianakiev, Oct 09 2017


STATUS

approved



