A129656 Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.

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

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.

Links

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

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.

Eric Weisstein's World of Mathematics, 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[ # ] &]
fun[p_, e_] := Module[{ b = IntegerDigits[e, 2]}, m=Length[b]; Product[If[b[[j]] > 0, 1+p^(2^(m-j)), 1], {j, 1, m}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[1000], isigma[#]>2# &] (* Amiram Eldar, May 12 2019 *)

Crossrefs

Cf. A126168, A049417, A127666, A129657, A007357.

Sequence in context: A292982 A334972 A109797 * A334974 A048945 A111398

Adjacent sequences: A129653 A129654 A129655 * A129657 A129658 A129659

Keyword

easy,nonn

Author

Ant King, Apr 29 2007

Status

approved