### Near Multiperfects

```                           Near-Multiperfects

A near-multiperfect has near-integer abundancy , i.e. , the sum of its
divisors leaves a small remainder when divided by some small multiple .
More specifically , a near-multiperfect is defined as any natural  n ,
with :

sum-of-divisors ( n )   =   k * n  +  r
abs ( r )   <=   loge ( n )

loge is the log to the base e = natural log = the Napierian logarithm .
This definition includes the multiperfect .
A proper near-multiperfect is a near-multiperfect which is not
multiperfect .
```

#### Least Near-Multiperfects

```This list of the least near-multiperfects excludes
a) primes ,
b) powers of 2 and
c) naturals of the form 6 times a prime .

n    r
6    0
10   -2
20    2
28    0
70    4
88    4
104    2
110   -4
120    0
136   -2
152   -4
464    2
496    0
592   -6
650    2
672    0
884   -4
1155   -6
1888    4
1952    2
2144   -4
4030    4
5830    4
8128    0
8384   -4
8925    6
11096    8
17816    8
18632   -4
18904   -8
30240    0
32128    4
32445    6
32760    0
32896   -2
33664   -8
45356    8
70564   -8
77744    8
85936   -8
91388    8
100804   -8
116624   -4
128768    8
130304    2
133376  -10
244036  -11
254012    8
388076    8
391612   -8
430272   12
442365    6
518656   10
521728    4
522752    2
523776    0
527872   -8
528896  -10
1090912   -8
1848964    4
2087936    8
2102272   -6
2178540    0
2291936    8
8378368    4
8382464    2
8394752   -4
9928792   16
11547352   16
12026888  -16
13174976    8
13192768  -16
15370304   -4
16102808  -16
17619844   -8
17999992   16
23569920    0
26347688  -16
29322008  -16
29465852    8
33501184   12
33550336    0
33653888  -16
35021696    8
45335936    8
45532800    0
73995392   -4
89283592   16
120888092    8
134094848   14
134193152    2
142990848    0
159030135   18
169371008  -16
173482552   16
173631608  -16
184773312   12
260378492    8
293947648  -16
361702144   16
381236216    8
459818240    0
536559616   18
536920064   -4
537051136  -12
537116672  -16
624032630   20
775397948    8
815634435   -6
883927808  -16
1081850752   16
1113445430   20
1379454720    0
1476304896    0
1550860550   20
1631268870  -18
1845991216   16
2146926592   16
2146992128   14
2147516416   -2
2147581952   -4
2147713024   -8
2147975168  -16
2493705728  -16
2586415095   18
3381872252    8
3915380170  -20
4856970752    8
5556840416  -16
6077111050  -20
6800228816    8
6800695312   -8
8589082624   12
8589344768    8
8589869056    0
9796360330  -20
10828121356  -20
11097907192   16
12985220152   16
13092865928  -16
14182439040    0
21818579968   16
31998395520    0
34356723712   22
34357510144   16
34358296576   10
34359083008    4
34360131584   -4
34360655872   -8
34360918016  -10
42783299288  -16

count = 151
The multiperfects have remainder , r = 0 .
This list is complete through
43279999099 .
```
The near-multiperfects are sequence A117349 in Neil J. A. Sloane's On-Line Encyclopedia of Integer Sequences .
```Walter Nissen
2008-01-21
```