OFFSET
1,2
COMMENTS
If p is prime and p == 2 (mod 3) then p^2 is in the sequence.
Let E(x) = #{n | a(n) <= x} be the number of terms of this sequence up to x. Kanold proved that there are two constants 0 < c1 < c2 and a positive number x_0 such that c1 < E(x)/sqrt(x/log(x)) < c2 for x > x_0. De Koninck and Kátai proved that there is a positive constant c such that E(x) = c * (1 + o(1)) * sqrt(x/log(x)).
Apparently most of the terms are squares or powers of 2. Terms that are not included 1250, 4802, 31250, 57122, ...
Numbers k such that A099377(k) = A038040(k) and A099378(k) = A000203(k). - Amiram Eldar, Nov 02 2021
REFERENCES
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 75.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Imre Kátai, On an estimate of Kanold, Int. J. Math. Anal., Vol. 5, No. 8 (2007), pp. 1-12.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl II, Mathematische Annalen, Vol. 134, No. 3 (1958), pp. 225-231.
MATHEMATICA
Select[Range[10^4], CoprimeQ[# * DivisorSigma[0, #], DivisorSigma[1, #]] &]
PROG
(Magma) [k:k in [1..5000]| Gcd(k*NumberOfDivisors(k), DivisorSigma(1, k)) eq 1]; // Marius A. Burtea, Dec 20 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Dec 20 2019
STATUS
approved