Solution of $\sigma(\sigma(m)+m)=2\sigma(m)$ with $\omega(m)>8$?

Question

This question is related to this one.

$\sigma(n)$ is the divisor-sum function (the sum of the positive divisors of $n$) and $\omega(n)$ is the number of distinct prime factors of $n$.

The object is to find a solution of the equation $$\sigma(\sigma(m)+m)=2\sigma(m)$$ with $\omega(m)>8$.

I know three solutions with $\omega(m)=8$ , namely

• $221\ 548\ 476\ 457$ with the prime factors $[7, 11, 17, 31, 37, 41, 59, 61]$
• $2\ 279\ 480\ 826\ 161$ with the prime factors $[7, 11, 23, 29, 41, 67, 107, 151]$
• $8\ 304\ 171\ 206\ 569$ with the prime factors $[7, 19, 29, 37, 71, 73, 103, 109]$

I tried all squarefree numbers with $9$ prime factors not exceeding the $30$ th prime factor without success. The desired number need not be squarefree , but I guess chances are best to find one if one concentrates on squarefree numbers.

Answer 1

I have found some solutions for $\omega(n) = 9$, they are

• $10\ 278\ 665\ 018\ 765$ with prime factors $[5, 11, 13, 17, 29, 31, 59, 107, 149]$.
• $601\ 610\ 156\ 911\ 127$ with prime factors $[7, 13, 37, 41, 53, 61, 79, 113, 151]$
• $212\ 579\ 943\ 696\ 197$ with prime factors $[7, 11, 31, 37, 41, 47, 79, 97, 163]$
• $23\ 036\ 114\ 773\ 555$ with prime factors $[5, 13, 17, 19, 29, 43, 47, 97, 193]$
• $75\ 567\ 210\ 262\ 915$ with prime factors $[5, 7, 19, 29, 31, 53, 89, 127, 211]$
• $2\ 416\ 836\ 034\ 112\ 071$ with prime factors $[11, 13, 43, 53, 59, 67, 89, 107, 197]$

Also, I have found additional $n$'s with $\omega(n)=8$, namely

• $18\ 880\ 317\ 877\ 057$ with prime factors $[7, 13, 23, 41, 89, 113, 131, 167]$
• $709\ 300\ 520\ 357$ with prime factors $[11, 13, 17, 23, 29, 31, 103, 137]$
• $94\ 948\ 236\ 621\ 139$ with prime factors $[11, 43, 53, 59, 61, 67, 113, 139]$
• $13\ 400\ 512\ 0551\ 179$ with prime factors $[19, 41, 43, 59, 67, 83, 89, 137]$
• Others are [2153102610835, 43933957290041, 26574181780289, 22507885465637, 161156016345533, 104945128479529, 165745991361431, 1029486795292891, 878769619506647, 2497072321919201, 288659619872767, 359622149388931, 56725558238701, 363106946743739, 1914827449993397, 529797038551483, 139935251361127, 2898629694746879, 6342940065472321, 326927610069689413, 14229497568359327, 1815589960085559887, 1628902320753074453].

A note is that I too only tested squarefree numbers, thus my results are naturally also squarefree. I tried to increase my search limit but my program overflows a 64-bit integer.

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Update: I found a non-squarefree solution with $\omega(n)=8$:

• 44573803189825 with prime factors $[5^2, 7, 13, 53, 97, 103, 163, 227]$

And some non-squarefree solutions with $\omega(n)=9$:

• 1123317615558775 with factors $[5^2, 7, 13, 17, 53, 71, 157, 211, 233]$
• 133912528729525 with factors $[5^2, 7, 13, 17, 31, 43, 113, 127, 181]$
• Examples with $7^2$ are 4104257232597193 and 11617802791571921