A117346 - OEIS

A117346

Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 70, 71, 73, 79, 83, 88, 89, 97, 101, 103, 104, 107, 109, 110, 113, 120, 127, 128, 131, 136, 137, 139, 149, 151, 152, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B2.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Multiperfect Number.

EXAMPLE

70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

MATHEMATICA

asmlQ[n_]:=Module[{p=Mod[DivisorSigma[1, n], n]}, If[p>n/2, p=n-p]; p<=Log[n]];

Select[Range[200], asmlQ] (* Harvey P. Dale, Dec 25 2013 *)

CROSSREFS

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117347 through A117350.