A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 37800, 42336, 43200, 48600, 50400, 54000, 56448, 57600, 59400, 63504, 64800, 70200, 72000, 75600, 79200, 81000, 84672, 86400, 88200, 90000, 91800, 93600, 97200, 98784, 100800, 102600

Offset: 1

Author

Amiram Eldar, Dec 28 2020

Keywords

• nonn

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Analogous to A068403 as A308053 is analogous to A005101.

Apparently, the least odd term in this sequence is 3^4 * 5^3 * 7^3 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29^2 = 3296233276111741840875.

The asymptotic density of this sequence is Sum_{n>=1} f(A364991(n)) = 0.0004006..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Aug 15 2023

Examples

			5400 is a term since csigma(5400) = 16380 > 3 * 5400.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A308053.

Cf. A007947, A057723, A364991 (primitive terms).

Similar sequences: A068403, A285615, A293187, A300664, A307112, A328135.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] > 3*# &]

PARI

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) > 3*n; \\ Amiram Eldar, Aug 15 2023