A187794 Sum of the perfect divisors of n.

0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 28, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 28, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 28, 0, 6

Offset: 1

Author

Timothy L. Tiffin, Jan 06 2013

Keywords

• nonn

Comments

Sum of divisors d of n with sigma(d) = 2*d.

Unless an odd perfect number exists, a(n) = 0 whenever n is odd.

If no odd perfect numbers exist: If n is 6 mod 12 then a(n) = 6, if n is 1, 2, 3, 5, 7, 9, 10, or 11 mod 12 then a(n) = 0.

The first occurrence of three nonzero values at three consecutive even indices is a(1984) = 496, a(1986) = 6, a(1988) = 28.

Records are at n = 6, 28, 84, 496, 1488, 3472, 8128, 24384, 56896, 170688, 251968, 755904, 1763776, 5291328, 33550336, 100651008, 234852352, 704557056, 1040060416, 3120181248, 4260892672, .... - Charles R Greathouse IV, Jan 16 2013

A185351 gives the distinct values of this sequence (or numbers in the range of the sum of perfect divisors function) in ascending order. - Timothy L. Tiffin, Jul 13 2016

Fixed points: a(n) = n if and only if n is a perfect number (A000396). - Timothy L. Tiffin, Jul 14 2016

Examples

			a(84) = 6+28 = 34 since both 6 and 28 divide 84. - _Timothy L. Tiffin_, Jul 14 2016
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000396, A185351.

Programs

Mathematica

a[n_] := DivisorSum[n, If[DivisorSigma[1, #] == 2#, #, 0]&]; Array[a, 114] (* Jean-François Alcover, Dec 18 2015 *)

PARI

a(n)=sumdiv(n,d,(sigma(d,-1)==2)*d) \\ Charles R Greathouse IV, Jan 16 2013