A064599 -id:A064599 - cp's OEIS frontend

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

Dean Hickerson, Sep 25 2001

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

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

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?

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

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

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 *)

A327944 Numbers m that are equal to the sum of their first k consecutive nonunitary divisors, but not all of them (i.e k < A048105(m)).

Original entry on oeis.org

480, 2688, 17640, 131712, 2095104, 3576000, 4248288, 16854816, 41055200, 400162032, 637787520, 788259840, 1839272960, 2423592576

Offset: 1

Amiram Eldar, Sep 30 2019

The nonunitary version of Erdős-Nicolas numbers (A194472).

If all the nonunitary divisors are permitted (i.e. k <= A048105(n)), then the nonunitary perfect numbers (A064591) are included.

			480 is in the sequence since its nonunitary divisors are 2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 40, 48, 60, 80, 120 and 240 and 2 + 4 + 6 + 8 + 10 + 12 + 16 + 20 + 24 + 30 + 40 + 48 + 60 + 80 + 120 = 480.~

Cf. A048105, A064591, A064597, A064599.

Cf. A194472, A293618, A309843.

ndivs[n_] := Block[{d = Divisors[n]}, Select[d, GCD[ #, n/# ] > 1 &]]; ndivs2[n_] := Module[{d=ndivs[n]},If[Length[d]<2,{},Drop[d, -1] ]]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n,ndivs2[n]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

cp's OEIS Frontend

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

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