This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069266 #23 Feb 11 2024 02:24:12
%S A069266 576,1296,2304,3600,5184,7056,8100,9216,11664,14400,17424,20736,24336,
%T A069266 28224,32400,36864,44100,46656,57600,63504,69696,72900,82944,90000,
%U A069266 97344,104976,108900,112896,129600,147456,152100,156816,166464,176400,186624,202500,207936
%N A069266 Numbers k such that Sum_{d|k} d/core(d) > 2*k, where core(d) is the squarefree part of d.
%C A069266 All terms <= 10^17 are squares. Are there any nonsquare terms? - _David A. Corneth_, Sep 05 2020
%C A069266 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 A069266 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 A069266 Numbers k such that A068976(k) > 2*k.
%e A069266 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 A069266 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 A069266 (PARI) is(n) = sumdiv(n, d, d/core(d)) > n << 1 \\ _David A. Corneth_, Sep 05 2020
%Y A069266 Cf. A068976, A007913.
%K A069266 nonn,easy
%O A069266 1,1
%A A069266 _Benoit Cloitre_, Apr 14 2002
%E A069266 More terms from _Amiram Eldar_, Sep 05 2020