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

Offset: 1

Author

Walter Nissen, Mar 09 2006

Keywords

• nonn

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.

Examples

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

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.

Crossrefs

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

Programs

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 *)

Extensions

First term prepended by Harvey P. Dale, Dec 25 2013