A341750 - OEIS

%I #12 Feb 21 2021 02:27:51

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,18,20,22,24,26,28,30,33,34,38,40,

%T 42,44,45,46,48,51,52,54,56,58,60,62,66,68,69,70,72,74,76,78,80,82,84,

%U 86,87,88,90,91,92,94,95,96,99,102,104,105,106,108,110,112

%N Numbers k such that gcd(k, sigma(k)) > log(log(k)).

%C Pollack (2011) proved that the asymptotic density of numbers k such that gcd(k, sigma(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).

%C There are only 10 terms of A014567 in this sequence: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13.

%H Amiram Eldar, <a href="/A341750/b341750.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Pollack, <a href="http://doi.org/10.1307/mmj/1301586311">On the greatest common divisor of a number and its sum of divisors</a>, Michigan Math. J., Vol. 60, No. 1 (2011), pp. 199-214.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dickman_function">Dickman function</a>.

%e 15 is a term since gcd(15, sigma(15)) = gcd(15, 24) = 3 > log(log(15)) = 0.996...

%e 16 is not a term since gcd(16, sigma(16)) = gcd(16, 31) = 1 < log(log(16)) = 1.0197...

%t Select[Range[100], GCD[#, DivisorSigma[1, #]] > Log[Log[#]] &]

%o (PARI) isok(k) = (k==1) || (gcd(k, sigma(k)) > log(log(k))); \\ _Michel Marcus_, Feb 20 2021

%Y Cf. A000203, A001620, A014567, A080130, A009194, A227242, A341749.

%K nonn

%O 1,2

%A _Amiram Eldar_, Feb 18 2021