I don't know what caused Visual Studio to start thinking it needs to publish my project, but now whenever I build it, I get several "Unable to apply publish properties for item 'blah'" warnings. How do I stop Visual&nbsp;Studio&nbsp;2005's publishing facilities from working on my project completely? I use InstallShield, so I don't have any use for Visual Studio's publishing system.


Question: Can we represent this sequence as a function of n?
Answer: I wrote out the factorizations of the first 24 numbers:

================================================================

#   A133411 (b-file synthesized from sequence entry)
                       2   3   5   7  11  13  17  19  23  29  31
                       -   -   -   -   -   -   -   -   -   -   -
1 1
2 2                    1
3 4                    2
4 12                   2   1
5 24                   3   1
6 48                   4   1
7 240                  4   1   1
8 720                  4   2   1
9 5040                 4   2   1   1
10 10080               5   2   1   1
11 20160               6   2   1   1
12 221760              6   2   1   1   1
13 665280              6   3   1   1   1
14 8648640             6   3   1   1   1   1
15 17297280            7   3   1   1   1   1
16 294053760           7   3   1   1   1   1   1
17 5587021440          7   3   1   1   1   1   1   1
18 27935107200         7   3   2   1   1   1   1   1
19 642507465600        7   3   2   1   1   1   1   1   1
20 1927522396800       7   4   2   1   1   1   1   1   1
21 13492656777600      7   4   2   2   1   1   1   1   1
22 26985313555200      8   4   2   2   1   1   1   1   1
23 782574093100800     8   4   2   2   1   1   1   1   1   1
24 24259796886124800   8   4   2   2   1   1   1   1   1   1   1
                       -   -   -   -   -   -   -   -   -   -   -
                       2   3   5   7  11  13  17  19  23  29  31


================================================================

There is no closed form expression for this. However, a more realistic goal is to have the ability to find the next larger element, $a_{n+1},$ in the list up to element $n,$ then the next one after that, and so on. This has been done. For general terminology, you want On Highly Composite Numbers by Jean-Louis Nicolas, pages 215-244 in a book called Ramanujan Revisited, edited by George Andrews et al, appearing in 1988. You may email me for a pdf. Then, the specific method is by his student Guy Robin, Methodes d'optimisation pour un probleme de theorie des nombres, R.A.I.R.O. Informatique theoretique 17, no. 3, 1983, pages 239-247.

The reason it is possible to do such a thing is that highly composite numbers bear some resemblance to superior highly composite numbers. S.H.C. numbers have the basic feature, nonincreasing exponents, but in addition the ratios of the exponents for distinct primes $p \neq q$ must approximate the outcome of Ramanujan's recipe.  Between consecutive S.H.C. numbers, one may find all the h.c. numbers as a sort of operations research problem. This is all, well, difficult. So it all depends on why this is important to you, and how important. The short version, meanwhile, is that you calculate the s.h.c larger than you current $a_n,$ calculate the h.c. numbers up that high, and take the smallest one that is a multiple of $a_n.$ If you need to go up an extra s.h.c number, so be it.

Note that finding the first s.h.c number which is a multiple of your fixed number $a_n$ is entirely straightforward, as for any given $p^e,$ one may find the first (largest) $\delta &gt; 0$ such that the s.h.c. number $N_\delta$ is divisible by $p^e.$ Do this for all primes that divide your fixed $a_n$ and there you have it.