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# Talk:Divisors

## Largely abundant numbers versus highly abundant numbers?

A?????? Largely abundant numbers: ${\displaystyle \scriptstyle \sigma (n)\,\geq \,\sigma (m)\,}$ for all ${\displaystyle \scriptstyle m\,<\,n\,}$.

{Is this the same sequence as A002093 (highly abundant numbers) or is the strong law of small numbers at play here? — Daniel Forgues 04:31, 23 May 2012 (UTC)}

A002093 Highly abundant numbers: ${\displaystyle \scriptstyle \sigma (n)\,>\,\sigma (m)\,}$ for all ${\displaystyle \scriptstyle m\,<\,n\,}$.

{1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, ...}

Compare with

A067128 Ramanujan's largely composite numbers, defined to be ${\displaystyle \scriptstyle n\,}$ such that ${\displaystyle \scriptstyle d(n)\,\geq \,d(k)\,}$ for ${\displaystyle \scriptstyle k\,=\,1\,}$ to ${\displaystyle \scriptstyle n-1\,}$.

{1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, ...}

A002182 Highly composite numbers, definition (1): where ${\displaystyle \scriptstyle d(n)\,}$, the number of divisors of ${\displaystyle \scriptstyle n\,}$ (A000005), increases to a record.

{1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, ...}

Daniel Forgues 04:33, 23 May 2012 (UTC)

They first differ at the 41st term since ${\displaystyle \sigma (660)=2016=\sigma (672)<\sigma (720).}$ Charles R Greathouse IV 05:17, 23 May 2012 (UTC)

Then

A?????? Largely abundant numbers: ${\displaystyle \scriptstyle \sigma (n)\,\geq \,\sigma (m)\,}$ for all ${\displaystyle \scriptstyle m\,<\,n\,}$.

{1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, ...}

doesn't seem to be in the OEIS. Do you want to create the sequence (since you saw that they were different... and I didn't!) — Daniel Forgues 07:07, 23 May 2012 (UTC)