Talk:Divisors

Largely abundant numbers versus highly abundant numbers?

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

{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: $\scriptstyle \sigma (n)\,>\,\sigma (m)\,$ for all $\scriptstyle m\,<\,n\,$.

{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 $\scriptstyle n\,$ such that $\scriptstyle d(n)\,\geq \,d(k)\,$ for $\scriptstyle k\,=\,1\,$ to $\scriptstyle n-1\,$.

{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 $\scriptstyle d(n)\,$, the number of divisors of $\scriptstyle n\,$ (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 $\sigma (660)=2016=\sigma (672)<\sigma (720).$ Charles R Greathouse IV 05:17, 23 May 2012 (UTC)

Then

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

{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)