Let σ(x) denote the sum of the divisors of the number x∈N, the set of positive integers. Denote the deficiency of x as D(x):=2x−σ(x).
This afternoon I noticed some interesting fact about the quantity n2/D(n2) where n2 is the non-Euler (i.e., square) part of members of the OEIS sequence A228059.
Specifically, we have the following values of n2 from members of OEIS sequence A228059 and the corresponding (apparently almost increasing, and non-integral) values of n2/D(n2):
n12=32=9,n12/D(n12)=9/5=1.8
n22=34=81,n22/D(n22)=81/41≈1.97561
n32=212=441,n32/D(n32)=147/47≈3.12766
n42=452=2025,n42/D(n42)=2025/299≈6.77258
n52=1352=18225,n52/D(n52)=18225/2567≈7.09973
n62=2852=81225,n62/D(n62)=27075/2969≈9.11923
n72=1652=27225,n72/D(n72)=27225/851≈31.9918
n82=7652=585225,n82/D(n82)=585225/18893≈30.9758
n92=76952=59213025,n92/D(n92)=19737675/731333≈26.9886.
Here is my question:
Is it always the case that n2/D(n2) is non-integral where n2 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⋅32
405=5⋅34
2205=5⋅(3⋅7)2
26325=13⋅(32⋅5)2
236925=13⋅(33⋅5)2
1380825=17⋅(3⋅5⋅19)2
1660725=61⋅(3⋅5⋅11)2
35698725=61⋅(32⋅5⋅17)2
3138290325=53⋅(34⋅5⋅19)2
29891138805=5⋅(32⋅112⋅71)2
73846750725=509⋅(3⋅5⋅11⋅73)2
194401220013=21557⋅(3⋅7⋅11⋅13)2
194509436121=21569⋅(3⋅7⋅11⋅13)2
194581580193=21577⋅(3⋅7⋅11⋅13)2
194689796301=21589⋅(3⋅7⋅11⋅13)2
194798012409=21601⋅(3⋅7⋅11⋅13)2
194906228517=21613⋅(3⋅7⋅11⋅13)2
194942300553=21617⋅(3⋅7⋅11⋅13)2
195230876841=21649⋅(3⋅7⋅11⋅13)2
195339092949=21661⋅(3⋅7⋅11⋅13)2
195447309057=21673⋅(3⋅7⋅11⋅13)2
195699813309=21701⋅(3⋅7⋅11⋅13)2
195808029417=21713⋅(3⋅7⋅11⋅13)2
196024461633=21737⋅(3⋅7⋅11⋅13)2
196204821813=21757⋅(3⋅7⋅11⋅13)2
196349109957=21773⋅(3⋅7⋅11⋅13)2
196745902353=21817⋅(3⋅7⋅11⋅13)2
196781974389=21821⋅(3⋅7⋅11⋅13)2
196962334569=21841⋅(3⋅7⋅11⋅13)2
197323054929=21881⋅(3⋅7⋅11⋅13)2
197431271037=21893⋅(3⋅7⋅11⋅13)2
197755919361=21929⋅(3⋅7⋅11⋅13)2
197828063433=21937⋅(3⋅7⋅11⋅13)2
198044495649=21961⋅(3⋅7⋅11⋅13)2
198188783793=21977⋅(3⋅7⋅11⋅13)2
198369143973=21997⋅(3⋅7⋅11⋅13)2
198513432117=22013⋅(3⋅7⋅11⋅13)2
(I used WolframAlpha for computing the prime factorizations of the 11th to 37th 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 (n2) of
(3⋅7⋅11⋅13)2
is deficient-perfect, and that this condition is known to be equivalent to the Descartes-Frenicle-Sorli conjecture that s=1, if qsn2 is an odd perfect number with special/Euler prime q. (That is, when
n2=(3⋅7⋅11⋅13)2
then
n2D(n2)=11011=7⋅112⋅13
is an integer.)
Lastly, by this answer, it is known that the Descartes spoof
D=32⋅72⋅112⋅132⋅22021=198585576189
is not a member of OEIS sequence A228059.
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