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%I #23 Feb 11 2024 02:24:12
%S 576,1296,2304,3600,5184,7056,8100,9216,11664,14400,17424,20736,24336,
%T 28224,32400,36864,44100,46656,57600,63504,69696,72900,82944,90000,
%U 97344,104976,108900,112896,129600,147456,152100,156816,166464,176400,186624,202500,207936
%N Numbers k such that Sum_{d|k} d/core(d) > 2*k, where core(d) is the squarefree part of d.
%C All terms <= 10^17 are squares. Are there any nonsquare terms? - _David A. Corneth_, Sep 05 2020
%C 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
%H David A. Corneth, <a href="/A069266/b069266.txt">Table of n, a(n) for n = 1..10000</a> (first 5000 terms from Amiram Eldar)
%F Numbers k such that A068976(k) > 2*k.
%e 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
%t 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 *)
%o (PARI) is(n) = sumdiv(n, d, d/core(d)) > n << 1 \\ _David A. Corneth_, Sep 05 2020
%Y Cf. A068976, A007913.
%K nonn,easy
%O 1,1
%A _Benoit Cloitre_, Apr 14 2002
%E More terms from _Amiram Eldar_, Sep 05 2020
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