A064591 - OEIS

%I #58 Mar 08 2024 09:02:37

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

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

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

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

%C 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

%C 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

%C 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

%C 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

%H Steve Ligh and Charles R. Wall, <a href="https://www.fq.math.ca/Scanned/25-4/ligh.pdf">Functions of Nonunitary Divisors</a>, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

%H MathOverflow, <a href="https://mathoverflow.net/questions/306760/">Are all nonunitary perfect numbers in the form 4k where k is an even perfect number</a>?

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

%t nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==n, Print[ n ] ] ]

%t 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 *)

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

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

%K nonn,hard,more

%O 1,1

%A _Dean Hickerson_, Sep 25 2001