Tuesday, 20 October 2015

number theory - On the quantity n2/D(n2) where n2 is the non-Euler part of members of the OEIS sequence A228059



Let σ(x) denote the sum of the divisors of the number xN, the set of positive integers. Denote the deficiency of x as D(x):=2xσ(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)=81/411.97561

n32=212=441,n32/D(n32)=147/473.12766

n42=452=2025,n42/D(n42)=2025/2996.77258

n52=1352=18225,n52/D(n52)=18225/25677.09973

n62=2852=81225,n62/D(n62)=27075/29699.11923

n72=1652=27225,n72/D(n72)=27225/85131.9918

n82=7652=585225,n82/D(n82)=585225/1889330.9758

n92=76952=59213025,n92/D(n92)=19737675/73133326.9886.



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=532


405=534

2205=5(37)2

26325=13(325)2

236925=13(335)2


1380825=17(3519)2

1660725=61(3511)2

35698725=61(32517)2

3138290325=53(34519)2

29891138805=5(3211271)2

73846750725=509(351173)2

194401220013=21557(371113)2

194509436121=21569(371113)2

194581580193=21577(371113)2

194689796301=21589(371113)2


194798012409=21601(371113)2

194906228517=21613(371113)2

194942300553=21617(371113)2

195230876841=21649(371113)2

195339092949=21661(371113)2

195447309057=21673(371113)2

195699813309=21701(371113)2

195808029417=21713(371113)2

196024461633=21737(371113)2

196204821813=21757(371113)2


196349109957=21773(371113)2

196745902353=21817(371113)2

196781974389=21821(371113)2

196962334569=21841(371113)2

197323054929=21881(371113)2

197431271037=21893(371113)2

197755919361=21929(371113)2

197828063433=21937(371113)2

198044495649=21961(371113)2

198188783793=21977(371113)2


198369143973=21997(371113)2

198513432117=22013(371113)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
(371113)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=(371113)2

then

n2D(n2)=11011=711213

is an integer.)



Lastly, by this answer, it is known that the Descartes spoof
D=327211213222021=198585576189


is not a member of OEIS sequence A228059.


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