[seqfan] Re: new sequences needing more terms

Hans Havermann gladhobo at teksavvy.com
Fri Jun 1 20:49:47 CEST 2012

a(n) = number of integers k >= 7 such that A212813(k) = n.
1, 3, 11, 2632

I wrote:

> Assuming that a(5) is indeed the sum of the number of prime  
> partitions of the 2632 numbers in a(4) doesn't just imply that "the  
> next term may be very large" (as Neil comments) but that a(5) is  
> essentially incalculable, since it would include the number of prime  
> partitions of 2*3^86093441-1. Is there even a way to approximate this?

I found my message to this list in the comment section of A212814, to  
which Neil added that there is an asymptotic formula for the sum of  
the number of prime partitions (which answers the "is there a way to  
approximate this" part of my query).

Unfortunately, having mistaken the offset in A212815, my use of  
2*3^86093441-1 was in error. To calculate a(5) of A212814 using my  
"sum of the number of prime partitions of one less than each of the  
2632 numbers in a(4)" assumption, one need only calculate the exact  
number of prime partitions of numbers up to 258280325. Having actually  
calculated these for numbers up to 50000 (a couple of years ago), I'm  
aware of the difficulty of the process. Scaling those results into the  
millions is not something that I could do on my antiquated hardware.  
Nonetheless, the much-smaller number places the notion of a(5) being  
"essentially incalculable" into one of "essentially do-able".

I'm going to edit/delete the latter part of the comment in A212814 to  
remove my mistake.

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