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 .
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