Smallest Odd Abundant Number not Divisible by 3

Theorem

The smallest odd abundant number not divisible by $3$ is $5 \, 391 \, 411 \, 025$.


Proof

We have:

$5 \, 391 \, 411 \, 025 = 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$

showing it is not divisible by $3$.


Then from $\sigma_1$ of $5 \, 391 \, 411 \, 025$ we have:

$\map {\sigma_1} {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800 = 2 \times 5 \, 391 \, 411 \, 025 + 16 \, 486 \, 750$

demonstrating that $5 \, 391 \, 411 \, 025$ is abundant.


This theorem requires a proof.
In particular: It is still to be shown it is the smallest such.
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Sources

  • 1990: M.T. Whalen and C.L. Miller: Odd abundant numbers: some interesting observations (J. Recr. Math. Vol. 22)
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5,391,411,025$
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