3 potentially new sequences

[N. J. A. Sloane]

seq(a3pr(2*n+1,false,false),n=1..nmax) ;
seq(a3pr(2*n+1,true,false),n=1..nmax) ;
seq(a3pr(n,false,false),n=1..nmax) ;
seq(a3pr(n,true,false),n=1..nmax) ;
seq(a3pr(2*n+1,false,true),n=1..nmax) ;
seq(a3pr(2*n+1,true,true),n=1..nmax) ;
seq(a3pr(n,false,true),n=1..nmax) ;
seq(a3pr(n,true,true),n=1..nmax) ;
seq(a2pr(2*n+1,2,false),n=1..nmax) ;
seq(a2pr(n,2,false),n=1..nmax) ;
seq(a2pr(2*n+1,2,true),n=1..nmax) ;
seq(a2pr(n,2,true),n=1..nmax) ;
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Subject: RE: 3 potentially new sequences
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A054860(n) = number of ways to write 2n+1 as the sum of three primes

a(n) = number of ways to write 2n as the sum of a prime and an
  almost prime
not found. sequence starts:
{0,0,0,1,1,0,2,1,2,1,2,2,2,3,5,1,4,2,3,6,5,3,5,5,3,4,7,4,7,6,4,10,6,4,10,6,6,7,9,5,9,9,5,8,11,4,11,10,10,13,9,7,10,10,10,10,9,8,11,13,5,16,14,9,15,11,11,13,15,13,13,10,11,15,17,8,19,11,11,17,15,15,17,18,10,13,17,15,14,14,11,19,14,13,21,18,16,17,18,14,19}

a(n) = number of ways to write 2n as the sum of two almost primes.
not found. start:
{0,0,0,0,1,1,1,1,1,2,2,0,2,1,2,3,2,1,3,1,3,3,4,1,5,3,3,2,3,2,6,1,5,4,5,3,7,3,4,4,5,2,7,4,5,6,6,3,7,4,8,5,7,4,9,5,7,3,8,3,12,3,7,7,9,5,10,5,7,5,9,7,13,6,10,8,9,8,12,5,9,7,11,5,15,9,11,5,12,8,19,6,12,6,11,6,14,8,9,10,12}

could someone please check the beginnings of th sequences?

Peter




> From seqfan-owner at ext.jussieu.fr  Mon Mar 10 11:33:59 2008
> From: petsie at dordos.net
> To: <seqfan at ext.jussieu.fr>
> Subject: RE: 3 potentially new sequences
> Cc: <njas at research.att.com>
> ... 
> a(n) = number of ways to write 2n as the sum of a prime and an
>   almost prime
> not found. sequence starts:
> {0,0,0,1,1,0,2,1,2,1,2,2,2,3,5,1,4,2,3,6,5,3,5,5,3,4,7,4,7,6,4,10,6,4,10,6,6,7,9,5,9,9,5,8,11,4,11,10,10,13,9,7,10,10,10,10,9,8,11,1
> 3,5,16,14,9,15,11,11,13,15,13,13,10,11,15,17,8,19,11,11,17,15,15,17,18,10,13,17,15,14,14,11,19,14,13,21,18,16,17,18,14,19}
> 
> a(n) = number of ways to write 2n as the sum of two almost primes.
> not found. start:
> {0,0,0,0,1,1,1,1,1,2,2,0,2,1,2,3,2,1,3,1,3,3,4,1,5,3,3,2,3,2,6,1,5,4,5,3,7,3,4,4,5,2,7,4,5,6,6,3,7,4,8,5,7,4,9,5,7,3,8,3,12,3,7,7,9,
> 5,10,5,7,5,9,7,13,6,10,8,9,8,12,5,9,7,11,5,15,9,11,5,12,8,19,6,12,6,11,6,14,8,9,10,12}
> 
> could someone please check the beginnings of th sequences?
>... 

These look alright to me. They are bi-sections of A072931 and A100949,

awk '{if(NR %2 ==0) printf "%d %d\n",$1/2,$2}' < b072931.txt > b072931_2n.txt
awk '{if(NR %2 ==0) printf "%d %d\n",$1/2,$2}' < b100949.txt > b100949_2n.txt

Richard
http://www.strw.leidenuniv.nl/~mathar