A341750 Numbers k such that gcd(k, sigma(k)) > log(log(k)).

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, 42, 44, 45, 46, 48, 51, 52, 54, 56, 58, 60, 62, 66, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 99, 102, 104, 105, 106, 108, 110, 112

Offset: 1

Author

Amiram Eldar, Feb 18 2021

Keywords

• nonn

Comments

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).

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

Examples

			15 is a term since gcd(15, sigma(15)) = gcd(15, 24) = 3 > log(log(15)) = 0.996...
16 is not a term since gcd(16, sigma(16)) = gcd(16, 31) = 1 < log(log(16)) = 1.0197...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J., Vol. 60, No. 1 (2011), pp. 199-214.
• Wikipedia, Dickman function.

Crossrefs

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

Programs

Mathematica

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

PARI

isok(k) = (k==1) || (gcd(k, sigma(k)) > log(log(k))); \\ Michel Marcus, Feb 20 2021