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

1,1

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.

Links

Table of n, a(n) for n=1..11.

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

Example

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.

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} ]

Crossrefs

{4} union 2*A000396.

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

Sequence in context: A009114 A144012 A197924 * A208940 A209068 A177265

Adjacent sequences: A065122 A065123 A065124 * A065126 A065127 A065128

Keyword

nonn

Author

Joseph L. Pe, Nov 13 2001

Extensions

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

Status

approved