A129575 Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.

900, 1764, 3600, 4356, 4500, 4900, 6084, 6300, 7056, 8100, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 25200, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 39600

Offset

1,1

Comments

There are only 52189 exponential abundant numbers less than 50 million, which suggests that these account for approximately 0.1% of all integers.

Includes 36*m for all m coprime to 6 that are not squarefree. - Robert Israel, Feb 19 2019

The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 0.001043673..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Peter Hagis, Jr., Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

Eric Weisstein's World of Mathematics, e-Divisor.

Example

The third integer that is exceeded by its proper exponential divisor sum is 3600. Hence a(3) = 3600.

Maple

filter:= proc(n)  local L, m, i, j;
  L:= ifactors(n)[2];
  m:= nops(L);
  mul(add(L[i][1]^j, j=numtheory:-divisors(L[i][2])), i=1..m)>2*n
end proc:
select(filter, [$1..10^5]); # Robert Israel, Feb 19 2019

Mathematica

ExponentialDivisors[1]={1}; ExponentialDivisors[n_]:=Module[{}, {pr, pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows], List]; Sort[Times@@(pr^Transpose[divpowers])]]; properexponentialdivisorsum[k_]:=Plus@@ExponentialDivisors[k]-k; Select[Range[5 10^4], properexponentialdivisorsum[ # ]># &]

Crossrefs

Cf. A126164, A051377, A049419, A054979, A054980, A328136.

Sequence in context: A318720 A338540 A137490 * A328136 A336254 A321206

Adjacent sequences: A129572 A129573 A129574 * A129576 A129577 A129578

Keyword

easy,nonn

Author

Ant King, Apr 28 2007

Status

approved