Talk:Highly composite number

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Favorable for being a highly composite number is having small prime factors such as 2 and 3, but not too many of the same: e.g. 16 has fewer divisors than 12.

Apr 11, 2003 14:32 Highly composite number (cur; hist) . . Tarquin (Talk) (fewer, not less -- and what does that sentence mean anyway?)

16=2x2x2x2, 12=2x2x3, factors 2 are more favorable than 3 (3x3x3x3 and 2x3x3 do not qualify), but many of the same are not. - Patrick 12:46 Apr 11, 2003 (UTC)


Yes, but is this a definition of a technical term "Favorable", or a rule of thumb for determining whether a number if HIghly Composite? it needs a reword either way -- Tarquin

I have rephrased it. - Patrick 16:39 Apr 11, 2003 (UTC)

Contents

[edit] 1 as a term of this sequence

NJA Sloane's sequence of highly composite numbers starts with 1. Anything illogical about starting this sequence with 1?? (Please do not confuse this with calling 1 a prime number, which is illogical because the properties of primes will fail.) 66.245.79.136 21:07, 5 Sep 2004 (UTC)

done by now. --FvdP 22:36, 21 Oct 2004 (UTC)

[edit] HCN connection to primes

my latest sequence to OEIS shows an undisputable connection of HCNs to primes as relates to the offset from each to the nearest primes. refer to McRae's site [1], who was nice enough to post the offset data. My conjecture that the absolute offset from each HCN to the nearest prime will itself always be prime holds thru at least the 384th HCN (credit to mensanator@aol.com for the extension). --Billymac00 14:57, 8 May 2006 (UTC)

Thank you for this information. I ask how many of these 160 HCNs is the difference greater than 9? If there are very few, then the conjecture is not yet credible, because 9 (the first possible non-prime difference) may have then been missed out by chance. Karl] 10:55, 3 May 2006

well, see [2] for my list. When >1, the (abs.) offset seems to be larger than the largest prime factor of the HCN ...a smaller example is 120=2**3 * 3 * 5 and the offset is 7. A larger example is HCN=27949074753600. Its factorization is 2**6 × 3**4 × 5**2 × 7 × 11 × 13 × 17 × 19 × 23 × 29 while the offset is 67. Anyway, looking at McRae's list of the 1st 120, about half of numbers are >1 and besides a solitary 7, all are >9. The sequence is loosely encountering primes in ascending order I think.--Billymac00 02:25, 5 May 2006 (UTC) corrected Karl 08:10, 5 May 2006 UT.

I've realised that the difference cannot have any common divisor with the HCN number, hence if it is not 1, then its smallest prime divisor of the difference must be larger than the largest prime divisor of the HCN. I think this along with some theorems about the spacing of primes may be used to prove the conjecture. Karl 08:30, 9 May 2006 UT.

[edit] Proven??

Has it been proven to be true that there is a highly composite number where all 4 of these numbers are composite??

  1. H-p
  2. H-1
  3. H+1
  4. H+p

where H is the number itself and p is the smallest prime not a factor of H. Georgia guy 17:14, 20 September 2006 (UTC)

The answer is yes. The smallest such number is 83160. The number p is equal to in this case is 13, because 83160 is divisible by 2, 3, 5, 7, and 11. So the numbers are:

  1. 83147 = 17 * 67 * 73
  2. 83159 = 137 * 607
  3. 83161 = 13 * 6397
  4. 83173 = 31 * 2683

Georgia guy 20:13, 20 September 2006 (UTC)

[edit] Harshad numbers?

The article currently states:

All highly composite numbers are also Harshad numbers.

without a mentioning a base. Saying "all bases" does not work, since according to that article, the only all-Harshad numbers are 1, 2, 4, and 6 - clearly only a tip of the iceberg of the highly composite numbers. Saying "some base" also does not work, since every number is a Harshad number in bases with a radix greater than or equal to itself, making the matter trivial. So what exactly is meant by this statement? -- Milo

It's true in bases 4 and 10 and any other bases where the digital root indicates divisibility by 3. It's probably not true in say, base 13. Thanks for pointing out this error. Anton Mravcek 23:25, 6 November 2006 (UTC)
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