|
| |
|
|
A129656
|
|
Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.
|
|
2
|
|
|
|
24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~7.95n. It follows that the density of the infinitary abundant numbers is 1/7.95, which is about 0.126.
|
|
|
REFERENCES
|
Cohen, Graeme L.; On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189, (1990), pp. 395-411.
|
|
|
LINKS
|
Table of n, a(n) for n=1..52.
Eric Weisstein's World of Mathematics, definition of infinitary divisor.
|
|
|
EXAMPLE
|
The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.
|
|
|
MATHEMATICA
|
ExponentList[n_Integer, factors_List]:={#, IntegerExponent[n, # ]}&/@factors; InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g]==g][ #, Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #, factors]&/@d]], _?(And@@#&), {1}]] ]] ] Null; properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k; InfinitaryAbundantNumberQ[k_]:=If[properinfinitarydivisorsum[k]>k, True, False]; Select[Range[500], InfinitaryAbundantNumberQ[ # ] &]
|
|
|
CROSSREFS
|
Cf. A126168, A049417, A127666, A129657, A007357..
Sequence in context: A098030 A068544 A109797 * A048945 A111398 A030626
Adjacent sequences: A129653 A129654 A129655 * A129657 A129658 A129659
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
Ant King, Apr 29 2007
|
|
|
STATUS
|
approved
|
| |
|
|
