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Numbers n such that the sums of the odd and even aliquot parts of n both divide n.
2

%I #10 Dec 26 2018 17:23:20

%S 4,12,56,992,16256,67100672,17179738112,274877382656,

%T 4611686016279904256,5316911983139663489309385231907684352,

%U 383123885216472214589586756168607276261994643096338432

%N Numbers n such that the sums of the odd and even aliquot parts of n both divide n.

%C I call the numbers in this sequence "just numbers", since they "fairly" allow both their odd and even aliquot part sums to divide them.

%C Vladeta Jovovic of Belgrade University proved that every just number not equal to 4 is twice a perfect number. The proof appears in the link.

%H Joseph L. Pe, <a href="http://www.numeratus.net/just/just.html">The Justice of Numbers: A Problem Proposal</a>, Nov 21 2001

%e The sum of the odd aliquot parts of 4 is 1 and the sum of the even aliquot parts of 4 is 2; both sums divide 4. The sum of the odd aliquot parts of 12 is 3 + 1 = 4 and the sum of the even aliquot parts of 12 is 6 + 4 + 2 = 12; both sums divide 12.

%t Do[d = Drop[ Divisors[n], -1]; l = Length[d]; ev = 0; od = 1; k = 2; While[k <= l, If[ EvenQ[ d[[k]]], ev = ev + d[[k]], od = od + d[[k]]]; k++ ]; If[ IntegerQ[n/ev] && IntegerQ[n/od], Print[n]], {n, 2, 10^6, 2} ]

%Y {4} union 2*A000396.

%Y Cf. A139256. [From _R. J. Mathar_, Nov 03 2008]

%K nonn

%O 1,1

%A _Joseph L. Pe_, Nov 13 2001

%E More terms from _Robert G. Wilson v_, Oct 10 2002