A065577 Number of Goldbach partitions of 10^n?

[zak seidov]

Richard,
thanks for your reply.
It seems this'd concern mainly Ivars!
Also could you please calculate cases n>=10) -
I guess this'd not take for you hours(days!)
as with my Mathematica.
thanks, Zak


--- Richard Mathar <mathar at strw.leidenuniv.nl> wrote:

> 
> zs> From seqfan-owner at ext.jussieu.fr  Wed Nov  1
> 06:44:53 2006
> zs> Return-Path: <seqfan-owner at ext.jussieu.fr>
> zs> Date: Tue, 31 Oct 2006 21:43:12 -0800 (PST)
> zs> From: zak seidov <zakseidov at yahoo.com>
> zs> Subject: A065577 Number of Goldbach partitions
> of 10^n? More terms?
> zs> To: seqfan at ext.jussieu.fr, ip at sciserv.org,
> rgwv at rgwv.com
> zs> 
> zs> Dear seqfans,
> zs> Bob, Ivars,
> zs> 
> zs> In A065577, 
> zs> a(1)=2 because 10=3+7=5+5?
> zs> Also a(2)=6 because
> zs> 100=3+97=11+89=17+83=29+71=41+59=47+53?
> zs> ....
> 
> One might call A065577 explicitly the partitions
> without regard to order. The
> conversion between the two counts is simple, see
> http://mathworld.wolfram.com/GoldbachPartition.html
> 
> The table of n, ordered, and non-ordered partitions
> is
> 
>                                     1, 3, 2
>                                    2, 12, 6
>                                    3, 56, 28
>                                   4, 254, 127
>                                  5, 1620, 810
>                                 6, 10804, 5402
>                                 7, 77614, 38807
> 
> with the ordered partitions in the OEIS as A073610,
> the non-ordered in A061358 (?)
> (Cf from A065577 to these two might also help to
> show the difference,
> A065577(n)=A061358(10^n).)
> With the exception of the n=1 leading term, the
> count of the
> ordered partitions is twice that of the non-ordered
> partitions,
> because 10^n/2 is not a prime then.
> 
> The Maple program to print the small table above:
> 
> A065577 := proc(n,orderd)
> 	local N,a,i,ip;
> 	N := 10^n ;
> 	a := 0 ;
> 	i := 1 ;
> 	ip := 2 ;
> 	while 2*ip <= N do
> 		if isprime(N-ip) then
> 			if orderd and ip <> N-ip then
> 				a := a+ 2;
> 			else
> 				a := a+ 1;
> 			fi ;
> 		fi ;
> 		i := i+1 ;
> 		ip := ithprime(i) ;
> 	od ;
> 	RETURN(a) ;
> end:
> 
> for n from 1 to 20 do
> 	print(n,A065577(n,true),A065577(n,false)) ;
> od ;
> 
> In PARI
> 
> A065577(n)={
> 	local(N,a,i,ip);
> 	N = 10^n ;
> 	a = 0 ;
> 	i = 1 ;
> 	ip = 2 ;
> 	while(2*ip <= N,
> 		if(isprime(N-ip),
> 			a++ ;
> 		) ;
> 		i++ ;
> 		\\ip = prime(i) ; \\ slower and needs -p switch
> 		ip = nextprime(ip+1) ; \\ pseudoprimes only
> 	) ;
> 	return(a) ;
> }
> 
> {
> 	for(n=1,20,
> 		print(n," ",A065577(n)) ;
> 	)
> }
> 
> The PARI output confirms the n=9 case quoted by Zak
> (I've not gone to n=10):
> 
> 1 2
> 2 6
> 3 28
> 4 127
> 5 810
> 6 5402
> 7 38807
> 8 291400
> 9 2274205
> 
> -- Richard
> 



 
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