A064591 Nonunitary perfect numbers: k is the sum of its nonunitary divisors; i.e., k = sigma(k) - usigma(k).

24, 112, 1984, 32512, 134201344, 34359476224, 549754765312

Offset: 1

Author

Dean Hickerson, Sep 25 2001

Keywords

• nonn
• hard
• more

Comments

There are no other terms up to 1.2*10^14.

If P (A000396) is an even perfect number, then 4*P is in the sequence. Are there any others?

If there are no terms of another form, the sequence goes on with 9223372032559808512 = 2^32 * A000668(8), 10633823966279326978618770463815368704 = 2^62 * A000668(9), 766247770432944429179173512337214552523989286192676864 = 2^90 * A000668(10), ... - Michel Lagneau, Jan 21 2015

Conjecture: let s0 be the sum of the inverses of the even divisors of a number n and s1 the sum of the inverses of the odd divisors of n; then n is in the sequence iff 2*s0-s1 = 1. - Michel Lagneau, Jan 21 2015

Ligh & Wall proved that 2^(p+1)*(2^p-1) is a term if p and 2^p-1 are primes, and that all the nonunitary perfect numbers below 10^6 are of this form. - Amiram Eldar, Sep 27 2018

If k is in the sequence and k = 2^m*p^a then k is of the form 4*P for an even perfect P. See the link to MathOverflow. - Joshua Zelinsky, Mar 07 2024

Examples

			The sum of the nonunitary divisors of 24 is 2 + 4 + 6 + 12 = 24.
		

Links

• Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
• MathOverflow, Are all nonunitary perfect numbers in the form 4k where k is an even perfect number?

Crossrefs

Cf. A000203 (sigma), A034448 (usigma), A048146, A063870.

Cf. A064592, A064593, A064594, A064595, A064596, A064597, A064598, A064599.

Programs

Mathematica

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==n, Print[ n ] ] ]
Do[s0=0;s1=0;Do[d=Divisors[n][[i]];If[Mod[d,2]==0,s0=s0+1/d,s1=s1+1/d],{i,1,Length[Divisors[n]]}];If[2*s0-s1==1,Print[n]],{n,2,10^9,2}] (* Michel Lagneau, Jan 21 2015 *)