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 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
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 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. LINKS Mathoverflow, Primes of the form a^2+1 EXAMPLE 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. 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: CROSSREFS Cf. A005574. Sequence in context: A252941 A069898 A245511 * A228829 A341982 A337686 Adjacent sequences:  A259937 A259938 A259939 * A259941 A259942 A259943 KEYWORD nonn AUTHOR Michel Lagneau, Jul 09 2015 STATUS approved

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Last modified September 20 17:42 EDT 2021. Contains 347588 sequences. (Running on oeis4.)