A065125 Numbers n such that the sums of the odd and even aliquot parts of n both divide n.

4, 12, 56, 992, 16256, 67100672, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432

Offset: 1

Author

Joseph L. Pe, Nov 13 2001

Keywords

• nonn

Comments

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

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.

Examples

			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.
		

Links

• Joseph L. Pe, The Justice of Numbers: A Problem Proposal, Nov 21 2001

Crossrefs

{4} union 2*A000396.

Cf. A139256. [From R. J. Mathar, Nov 03 2008]

Programs

Mathematica

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} ]

Extensions

More terms from Robert G. Wilson v, Oct 10 2002