A065577 Number of Goldbach partitions of 10^n?

[Richard Mathar]

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