OFFSET
1,1
COMMENTS
Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025
Analogous to 3-nondeficient numbers (A023197). Also, exponential 3-abundant numbers (analogous to A068403), if there are no exponential 3-perfect numbers. - Amiram Eldar, Sep 16 2025
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
W. Aiello, G. E. Hardy, and M. V. Subbarao, On the existence of e-multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 65-71.
MATHEMATICA
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]
PROG
(PARI) esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d)); }
isok(k) = esigma(k) >= 3*k; \\ Amiram Eldar, Sep 16 2025
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 04 2019
EXTENSIONS
Name corrected by Amiram Eldar, Sep 16 2025
STATUS
approved