OFFSET
1,5
COMMENTS
There cannot be a solution for an even number of terms on the l.h.s. because there would be an odd number of odd terms but the r.h.s. is even.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..300
FORMULA
a(n) = [x^4] Product_{k=2..2*n-1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024
EXAMPLE
a(1) = a(2) = 0 because prime(1) and prime(1) +- prime(2) +- prime(3) is always different from -2.
a(3) = 1 because prime(1) - prime(2) - prime(3) - prime(4) + prime(5) = -2.
MAPLE
s:= proc(n) option remember;
`if`(n<2, 0, ithprime(n)+s(n-1))
end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=1, 1,
b(abs(n-ithprime(i)), i-1)+b(n+ithprime(i), i-1)))
end:
a:= n-> b(4, 2*n-1):
seq(a(n), n=1..30); # Alois P. Heinz, Aug 08 2015
MATHEMATICA
s[n_] := s[n] = If[n<2, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 1, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[4, 2*n-1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
PROG
(PARI) A261057(n, rhs=-2, firstprime=1)={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.
(PARI) a(n, s=-2-prime(1), p=1)={if(n<=s, if(s==p, n==s, a(abs(n-p), s-p, precprime(p-1))+a(n+p, s-p, precprime(p-1))), if(s<=0, a(abs(s), max(sum(i=p+1, p+=2*n-2+bittest(s, 0), prime(i)), 1), prime(p))))} \\ M. F. Hasler, Aug 09 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Aug 08 2015
EXTENSIONS
a(26)-a(30) from Alois P. Heinz, Jan 04 2019
STATUS
approved