OFFSET
1,5
COMMENTS
There cannot be a solution for an even number of terms on the l.h.s. because all terms are odd but the r.h.s. is odd, too.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..300
FORMULA
a(n) = [x^6] Product_{k=4..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024
EXAMPLE
a(1) = a(2) = 0 because prime(3) and prime(3) +- prime(4) +- prime(5) are different from -1 for any choice of the signs.
a(3) = 0 because the same sums prime(3) +- ... +- prime(7) is also always different from -1 for any choice of the signs.
a(4) = 1 because prime(3) - prime(4) - prime(5) - prime(6) - prime(7) + prime(8) + prime(9) = -1 is the only solution.
MAPLE
s:= proc(n) option remember;
`if`(n<4, 0, ithprime(n)+s(n-1))
end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=3, 1,
b(abs(n-ithprime(i)), i-1)+b(n+ithprime(i), i-1)))
end:
a:= n-> b(6, 2*n+1):
seq(a(n), n=1..30); # Alois P. Heinz, Aug 08 2015
MATHEMATICA
s[n_] := s[n] = If[n<4, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 3, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[6, 2*n+1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
PROG
(PARI) A261063(n, rhs=-1, firstprime=3)={rhs-=prime(firstprime); my(p=vector(2*n-2+bittest(rhs, 0), i, prime(i+firstprime))); sum(i=1, 2^#p-1, sum(j=1, #p, (-1)^bittest(i, j-1)*p[j])==rhs)} \\ For illustrative purpose; too slow for n >> 10.
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Aug 08 2015
EXTENSIONS
a(15)-a(30) from Alois P. Heinz, Aug 08 2015
STATUS
approved