A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

120, 840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3480, 3720, 4440, 4920, 5160, 5640, 5880, 6360, 7080, 7320, 7560, 8040, 8520, 8760, 9240, 9480, 9720, 9960, 10680, 10920, 11640, 11880, 12120, 12360

Offset: 1

Author

Amiram Eldar, Mar 10 2018

Keywords

• nonn

Comments

Analogous to 3-abundant numbers (A023197) with isigma (A049417) instead of sigma (A000203).

Examples

			840 is in the sequence since isigma(840) = 2880 > 3 * 840.
		

Links

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

Crossrefs

Cf. A007357, A023197, A049417, A129656, A129657.

Programs

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}]]]]]; properinfinitarydivisorsum[k] := Plus @@ InfinitaryDivisors[k] - k; Infinitary3AbundantNumberQ[k_] :=  If[properinfinitarydivisorsum[k] >= 2 k, True, False]; Select[Range[15000], Infinitary3AbundantNumberQ[#] &] (* after Ant King at A129656 *)