A259940 Let A={A005574(n)}, the set of all numbers n for which n^2+1 is prime. The sequence lists the number of decompositions A005574(n) = A005574(n1) + A005574(n2) for some n1, n2 and every A005574(n)>1.

0, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 4, 1, 3, 2, 1, 1, 4, 4, 5, 2, 5, 3, 5, 8, 5, 5, 8, 6, 7, 7, 6, 7, 6, 6, 5, 8, 7, 8, 7, 11, 12, 6, 12, 8, 11, 12, 8, 11, 9, 8, 10, 13, 11, 6, 10, 8, 12, 11, 13, 12, 10, 17, 9, 8, 10, 13, 11, 15, 11, 9, 8, 14, 13, 12, 8, 8, 7, 9, 7

Offset: 1

Author

Michel Lagneau, Jul 09 2015

Keywords

• nonn

Comments

We use a little-known conjecture by Goldbach on the primes of form n^2+1: let A be the set of all numbers a for which a^2+1 is prime (A={1, 2, 4, 6, 10, ...}). Then every a in A (a>1) can be written in the form a=b+c for b,c in A.

Examples

			a(20)=5 because A005574(20)= 110 =>
A005574(20)= A005574(7)  + A005574(19)= 16 + 94,
A005574(20)= A005574(8)  + A005574(18)= 20 + 90,
A005574(20)= A005574(10) + A005574(17)= 26 + 84,
A005574(20)= A005574(11) + A005574(16)= 36 + 74,
A005574(20)= A005574(13) + A005574(14)= 54 + 56, for a total of five decompositions.
		

Links

• Mathoverflow, Primes of the form a^2+1

Crossrefs

Cf. A005574.

Programs

Maple

T:=array(1..112):
nn:=1000:k:=0:
for i from 1 to nn do:
p:=i^2+1:if type(p,prime)=true
then
k:=k+1:T[k]:=i:
else fi:
od:
for n from 1 to k do:q:=T[n]:it:=0:
  for a from 1 to k do:p1:=T[a]:
   for b from a to k do:p2:=T[b]:
    if q=p1+p2
    then
    it:=it+1:
    else fi:
   od:
od:
printf(`%d, `,it):
od: