A069266 Numbers k such that Sum_{d|k} d/core(d) > 2*k, where core(d) is the squarefree part of d.

576, 1296, 2304, 3600, 5184, 7056, 8100, 9216, 11664, 14400, 17424, 20736, 24336, 28224, 32400, 36864, 44100, 46656, 57600, 63504, 69696, 72900, 82944, 90000, 97344, 104976, 108900, 112896, 129600, 147456, 152100, 156816, 166464, 176400, 186624, 202500, 207936

Offset: 1

Author

Benoit Cloitre, Apr 14 2002

Keywords

• nonn
• easy

Comments

All terms <= 10^17 are squares. Are there any nonsquare terms? - David A. Corneth, Sep 05 2020

All the terms are squares. Proof: Let f(n) = A068976(n)/n. f(n) is multiplicative with f(p^e) = (p^2 + 1 - 2/p^e)/(p^2-1) if e is even and 2*(p - 1/p^e)/(p^2-1) if e is odd. Both are strictly increasing with e, the limits as e -> oo are f_even(p) = (p^2+1)/(p^2-1) and f_odd(p) = 2*p/(p^2-1), respectively, and f_odd(p) < f_even(p) for all primes p. The upper bound on f(n) is being attained at even exponents: f(n) < lim_{e->oo} Product_{p prime} (p^2 + 1 - 2/p^e)/(p^2-1) = Product_{p prime} f_even(p) = 5/2. If k is not a square, then there is at least one prime q|k with an odd exponent. Replacing the factor f_even(q) with f_odd(q) in the infinite product, we get f(k) < (5/2) * f_odd(q)/f_even(q) = 5*q/(q^2+1) <= 2. Therefore, A068976(k) = f(k) * k < 2*k and k is not a term. - Amiram Eldar, Feb 11 2024

Examples

			3600 is in the sequence as its divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, ..., 400, 450, 600, 720, 900, 1200, 1800, 3600 and the respective values d/core(d) are 1, 1, 1, 4, 1, 1, 4, 9, ... , 900, 400, 900, 3600. The sum of this latter list is 7722 which is greater than 2*3600. - _David A. Corneth_, Sep 05 2020
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5000 terms from Amiram Eldar)

Crossrefs

Cf. A068976, A007913.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], 2*(p^(e + 1) - 1)/(p^2 - 1), (p^(e + 2) + p^e - 2)/(p^2 - 1)]; Select[Range[2, 150000], Times @@ (f @@@ FactorInteger[#]) > 2*# &] (* Amiram Eldar, Sep 05 2020 *)

PARI

is(n) = sumdiv(n, d, d/core(d)) > n << 1 \\ David A. Corneth, Sep 05 2020

Formula

Numbers k such that A068976(k) > 2*k.

Extensions

More terms from Amiram Eldar, Sep 05 2020