Let $\sigma(x)$ denote the sum of the divisors of the number $x \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $x$ as $D(x):=2x-\sigma(x)$.
This afternoon I noticed some interesting fact about the quantity ${n^2}/D(n^2)$ where $n^2$ is the non-Euler (i.e., square) part of members of the OEIS sequence A228059.
Specifically, we have the following values of $n^2$ from members of OEIS sequence A228059 and the corresponding (apparently almost increasing, and non-integral) values of ${n^2}/D(n^2)$:
$${n_1}^2 = 3^2 = 9, {{n_1}^2}/D({n_1}^2) = 9/5 = 1.8$$
$${n_2}^2 = 3^4 = 81, {{n_2}^2}/D({n_2}^2) = {81}/{41} \approx 1.97561$$
$${n_3}^2 = {21}^2 = 441, {{n_3}^2}/D({n_3}^2) = {147}/{47} \approx 3.12766$$
$${n_4}^2 = {45}^2 = 2025, {{n_4}^2}/D({n_4}^2) = {2025}/{299} \approx 6.77258$$
$${n_5}^2 = {135}^2 = 18225, {{n_5}^2}/D({n_5}^2) = {18225}/{2567} \approx 7.09973$$
$${n_6}^2 = {285}^2 = 81225, {{n_6}^2}/D({n_6}^2) = {27075}/{2969} \approx 9.11923$$
$${n_7}^2 = {165}^2 = 27225, {{n_7}^2}/D({n_7}^2) = {27225}/{851} \approx 31.9918$$
$${n_8}^2 = {765}^2 = 585225, {{n_8}^2}/D({n_8}^2) = {585225}/{18893} \approx 30.9758$$
$${n_9}^2 = {7695}^2 = 59213025, {{n_9}^2}/D({n_9}^2) = {19737675}/{731333} \approx 26.9886.$$
Here is my question:
Is it always the case that ${n^2}/D(n^2)$ is non-integral where $n^2$ is the non-Euler (i.e., square) part of members of the OEIS sequence A228059?
Updated August 28 2018
The short answer to my original question is NO. (See the answer below.)
I ask because it is known that the exponent of the special / Euler prime of an odd perfect number is $1$ if and only if the non-Euler part is deficient-perfect. Coincidentally, in OEIS sequence A228059, all of the special / Euler primes for the first $9$ terms have exponent $1$.
Answer
Here are the first $37$ terms of the OEIS sequence A228059:
$$45 = 5\cdot{3^2}$$
$$405 = 5\cdot{3^4}$$
$$2205 = 5\cdot(3\cdot7)^2$$
$$26325 = 13\cdot({3^2}\cdot5)^2$$
$$236925 = 13\cdot({3^3}\cdot5)^2$$
$$1380825 = 17\cdot(3\cdot5\cdot19)^2$$
$$1660725 = 61\cdot(3\cdot5\cdot11)^2$$
$$35698725 = 61\cdot({3^2}\cdot5\cdot17)^2$$
$$3138290325 = 53\cdot({3^4}\cdot5\cdot19)^2$$
$$29891138805 = {5}\cdot({3^2}\cdot{{11}^2}\cdot{71})^2$$
$$73846750725 = {509}\cdot({3}\cdot{5}\cdot{11}\cdot{73})^2$$
$$194401220013 = {21557}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$194509436121 = {21569}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$194581580193 = {21577}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$194689796301 = {21589}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$194798012409 = {21601}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$194906228517 = {21613}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$194942300553 = {21617}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$195230876841 = {21649}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$195339092949 = {21661}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$195447309057 = {21673}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$195699813309 = {21701}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$195808029417 = {21713}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$196024461633 = {21737}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$196204821813 = {21757}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$196349109957 = {21773}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$196745902353 = {21817}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$196781974389 = {21821}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$196962334569 = {21841}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$197323054929 = {21881}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$197431271037 = {21893}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$197755919361 = {21929}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$197828063433 = {21937}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$198044495649 = {21961}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$198188783793 = {21977}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$198369143973 = {21997}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
$$198513432117 = {22013}\cdot({3}\cdot{7}\cdot{11}\cdot{13})^2$$
(I used WolframAlpha for computing the prime factorizations of the $11$th to $37$th terms.) Note that each of the first $37$ terms of OEIS sequence A228059 have a $p$ with exponent $1$.
Furthermore, note that the non-Euler part value ($n^2$) of
$$({3}\cdot{7}\cdot{11}\cdot{13})^2$$
is deficient-perfect, and that this condition is known to be equivalent to the Descartes-Frenicle-Sorli conjecture that $s=1$, if $q^s n^2$ is an odd perfect number with special/Euler prime $q$. (That is, when
$$n^2 = ({3}\cdot{7}\cdot{11}\cdot{13})^2$$
then
$$\frac{n^2}{D(n^2)} = 11011 = {7}\cdot{{11}^2}\cdot{13}$$
is an integer.)
Lastly, by this answer, it is known that the Descartes spoof
$$\mathscr{D} = {3^2}\cdot{7^2}\cdot{{11}^2}\cdot{{13}^2}\cdot{22021} = 198585576189$$
is not a member of OEIS sequence A228059.
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