Vtyjkyuk

Do all terms $n$ of OEIS sequence A228059 have a $p$ with exponent $1$?

OEIS sequence A228059: Odd numbers of the form $p^1+4kr^2$, where $p$ is prime of the form $1+4m$, $r > 1$, and $gcd(p,r) = 1$ that are closer to being perfect than previous terms.

Here are the first couple of terms:
$$45 = 5cdot3^2$$
$$405 = 5cdot3^4$$
$$2205 = 5cdot(3cdot7)^2$$
$$26325 = 13cdot(3^2cdot5)^2$$
$$236925 = 13cdot(3^3cdot5)^2$$
$$1380825 = 17cdot(3cdot5cdot19)^2$$
$$1660725 = 61cdot(3cdot5cdot11)^2$$
$$35698725 = 61cdot(3^2cdot5cdot17)^2$$
$$3138290325 = 53cdot(3^4cdot5cdot19)^2$$

UPDATE - July 05 2017 (7:30 PM - Manila time): I am currently running the Mathematica code referenced in the OEIS sequence to compute more terms past $3138290325$.

UPDATE - July 06 2017 (1:00 AM - Manila time): The Mathematica code is still running and is currently at $35698725$. It has not displayed $3138290325$ yet.

Here are the first $37$ terms:

$$45 = 5cdot3^2$$
$$405 = 5cdot3^4$$
$$2205 = 5cdot(3cdot7)^2$$
$$26325 = 13cdot(3^2cdot5)^2$$
$$236925 = 13cdot(3^3cdot5)^2$$
$$1380825 = 17cdot(3cdot5cdot19)^2$$
$$1660725 = 61cdot(3cdot5cdot11)^2$$
$$35698725 = 61cdot(3^2cdot5cdot17)^2$$
$$3138290325 = 53cdot(3^4cdot5cdot19)^2$$
$$29891138805 = 5cdot(3^2cdot11^2cdot71)^2$$
$$73846750725 = 509cdot(3cdot5cdot11cdot73)^2$$
$$194401220013 = 21557cdot(3cdot7cdot11cdot13)^2$$
$$194509436121 = 21569cdot(3cdot7cdot11cdot13)^2$$
$$194581580193 = 21577cdot(3cdot7cdot11cdot13)^2$$
$$194689796301 = 21589cdot(3cdot7cdot11cdot13)^2$$
$$194798012409 = 21601cdot(3cdot7cdot11cdot13)^2$$
$$194906228517 = 21613cdot(3cdot7cdot11cdot13)^2$$
$$194942300553 = 21617cdot(3cdot7cdot11cdot13)^2$$
$$195230876841 = 21649cdot(3cdot7cdot11cdot13)^2$$
$$195339092949 = 21661cdot(3cdot7cdot11cdot13)^2$$
$$195447309057 = 21673cdot(3cdot7cdot11cdot13)^2$$
$$195699813309 = 21701cdot(3cdot7cdot11cdot13)^2$$
$$195808029417 = 21713cdot(3cdot7cdot11cdot13)^2$$
$$196024461633 = 21737cdot(3cdot7cdot11cdot13)^2$$
$$196204821813 = 21757cdot(3cdot7cdot11cdot13)^2$$
$$196349109957 = 21773cdot(3cdot7cdot11cdot13)^2$$
$$196745902353 = 21817cdot(3cdot7cdot11cdot13)^2$$
$$196781974389 = 21821cdot(3cdot7cdot11cdot13)^2$$
$$196962334569 = 21841cdot(3cdot7cdot11cdot13)^2$$
$$197323054929 = 21881cdot(3cdot7cdot11cdot13)^2$$
$$197431271037 = 21893cdot(3cdot7cdot11cdot13)^2$$
$$197755919361 = 21929cdot(3cdot7cdot11cdot13)^2$$
$$197828063433 = 21937cdot(3cdot7cdot11cdot13)^2$$
$$198044495649 = 21961cdot(3cdot7cdot11cdot13)^2$$
$$198188783793 = 21977cdot(3cdot7cdot11cdot13)^2$$
$$198369143973 = 21997cdot(3cdot7cdot11cdot13)^2$$
$$198513432117 = 22013cdot(3cdot7cdot11cdot13)^2$$

(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
$$(3cdot7cdot11cdot13)^2$$
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 $q$.

Lastly, by this answer, it is known that the Descartes spoof
$$mathscrD = 3^2cdot7^2cdot11^2cdot13^2cdot22021 = 198585576189$$
is not a member of OEIS sequence A228059.