Specific evidence for or against the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

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This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd perfect number with special / Euler prime $q$ (satisfying $q equiv k equiv 1 pmod 4$ and $gcd(q,n)=1$). (Specifically, via exhaustion of all possible cases, it appears plausible that $k>1$ must hold true.)



On the other hand, for the Descartes spoof (which is probably the closest we will ever get to an "actual odd perfect number")
$$mathscrD = 3^2cdot7^2cdot11^2cdot13^2cdot22021 = 198585576189$$
the quasi-Euler prime $22021$ (which is actually composite) has exponent $1$. Additionally, the non-Euler part $3^2cdot7^2cdot11^2cdot13^2$ of $mathscrD$ can be (directly) shown to be deficient-perfect, a condition which is known to be equivalent to the Descartes-Frenicle-Sorli conjecture.



So here is my question:




What other evidence or heuristic do you know of that supports (or otherwise does not support) the Descartes-Frenicle-Sorli conjecture for odd perfect numbers?




One piece of evidence/heuristic per answer only, please.




elementary-number-theory big-list conjectures divisor-sum perfect-numbers




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    This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd perfect number with special / Euler prime $q$ (satisfying $q equiv k equiv 1 pmod 4$ and $gcd(q,n)=1$). (Specifically, via exhaustion of all possible cases, it appears plausible that $k>1$ must hold true.)



    On the other hand, for the Descartes spoof (which is probably the closest we will ever get to an "actual odd perfect number")
    $$mathscrD = 3^2cdot7^2cdot11^2cdot13^2cdot22021 = 198585576189$$
    the quasi-Euler prime $22021$ (which is actually composite) has exponent $1$. Additionally, the non-Euler part $3^2cdot7^2cdot11^2cdot13^2$ of $mathscrD$ can be (directly) shown to be deficient-perfect, a condition which is known to be equivalent to the Descartes-Frenicle-Sorli conjecture.



    So here is my question:




    What other evidence or heuristic do you know of that supports (or otherwise does not support) the Descartes-Frenicle-Sorli conjecture for odd perfect numbers?




    One piece of evidence/heuristic per answer only, please.




    elementary-number-theory big-list conjectures divisor-sum perfect-numbers




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      up vote
      1
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      up vote
      1
      down vote

      favorite











      This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd perfect number with special / Euler prime $q$ (satisfying $q equiv k equiv 1 pmod 4$ and $gcd(q,n)=1$). (Specifically, via exhaustion of all possible cases, it appears plausible that $k>1$ must hold true.)



      On the other hand, for the Descartes spoof (which is probably the closest we will ever get to an "actual odd perfect number")
      $$mathscrD = 3^2cdot7^2cdot11^2cdot13^2cdot22021 = 198585576189$$
      the quasi-Euler prime $22021$ (which is actually composite) has exponent $1$. Additionally, the non-Euler part $3^2cdot7^2cdot11^2cdot13^2$ of $mathscrD$ can be (directly) shown to be deficient-perfect, a condition which is known to be equivalent to the Descartes-Frenicle-Sorli conjecture.



      So here is my question:




      What other evidence or heuristic do you know of that supports (or otherwise does not support) the Descartes-Frenicle-Sorli conjecture for odd perfect numbers?




      One piece of evidence/heuristic per answer only, please.




      elementary-number-theory big-list conjectures divisor-sum perfect-numbers




      share|cite|improve this question













      This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd perfect number with special / Euler prime $q$ (satisfying $q equiv k equiv 1 pmod 4$ and $gcd(q,n)=1$). (Specifically, via exhaustion of all possible cases, it appears plausible that $k>1$ must hold true.)



      On the other hand, for the Descartes spoof (which is probably the closest we will ever get to an "actual odd perfect number")
      $$mathscrD = 3^2cdot7^2cdot11^2cdot13^2cdot22021 = 198585576189$$
      the quasi-Euler prime $22021$ (which is actually composite) has exponent $1$. Additionally, the non-Euler part $3^2cdot7^2cdot11^2cdot13^2$ of $mathscrD$ can be (directly) shown to be deficient-perfect, a condition which is known to be equivalent to the Descartes-Frenicle-Sorli conjecture.



      So here is my question:




      What other evidence or heuristic do you know of that supports (or otherwise does not support) the Descartes-Frenicle-Sorli conjecture for odd perfect numbers?




      One piece of evidence/heuristic per answer only, please.





      elementary-number-theory big-list conjectures divisor-sum perfect-numbers





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      asked Aug 2 at 12:46









      Jose Arnaldo Bebita Dris

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