A330606 - OEIS

%I #39 Sep 08 2022 08:46:24

%S 1,2,4,8,9,16,25,36,64,81,100,121,128,144,225,256,289,324,400,484,512,

%T 529,576,625,729,841,900,1024,1089,1156,1250,1296,1600,1681,1936,2025,

%U 2116,2209,2304,2401,2500,2601,2809,3025,3364,3481,3600,4096,4356,4624,4761

%N Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

%C If p is prime and p == 2 (mod 3) then p^2 is in the sequence.

%C 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)).

%C Apparently most of the terms are squares or powers of 2. Terms that are not included 1250, 4802, 31250, 57122, ...

%C Numbers k such that A099377(k) = A038040(k) and A099378(k) = A000203(k). - _Amiram Eldar_, Nov 02 2021

%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 75.

%H Amiram Eldar, <a href="/A330606/b330606.txt">Table of n, a(n) for n = 1..10000</a>

%H Jean-Marie De Koninck and Imre Kátai, <a href="http://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/jmdk/Documents/Publications/2007_on_a_estimate_of_kanold.pdf">On an estimate of Kanold</a>, Int. J. Math. Anal., Vol. 5, No. 8 (2007), pp. 1-12.

%H Hans-Joachim Kanold, <a href="https://doi.org/10.1007/BF01343208">Über das harmonische Mittel der Teiler einer natürlichen Zahl II</a>, Mathematische Annalen, Vol. 134, No. 3 (1958), pp. 225-231.

%t Select[Range[10^4], CoprimeQ[# * DivisorSigma[0, #], DivisorSigma[1, #]] &]

%o (Magma) [k:k in [1..5000]| Gcd(k*NumberOfDivisors(k),DivisorSigma(1,k)) eq 1]; // _Marius A. Burtea_, Dec 20 2019

%Y Cf. A000005, A000203, A038040, A099377, A099378, A324121.

%Y Subsequence of A014567 and A046678.

%K nonn

%O 1,2

%A _Amiram Eldar_, Dec 20 2019