OFFSET
1,3
COMMENTS
There cannot be a solution for an odd number of terms on the l.h.s. because there would be an even number of odd terms but the r.h.s. is odd.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..300
FORMULA
Conjecture: limit_{n->infinity} a(n)^(1/n) = 4. - Vaclav Kotesovec, Jun 05 2019
a(n) = [x^3] Product_{k=2..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024
EXAMPLE
a(1) = 1 counts the solution prime(1) - prime(2) = -1.
a(2) = 0 because prime(1) +- prime(2) +- prime(3) +- prime(4) is always different from -1.
a(3) = 2 counts the two solutions prime(1) - prime(2) + prime(3) - prime(4) - prime(5) + prime(6) = -1 and prime(1) - prime(2) - prime(3) + prime(4) + prime(5) - prime(6) = -1.
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(3, 2*n):
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[3, 2*n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
PROG
(PARI) A261061(n, rhs=-1, 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.
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Aug 08 2015
EXTENSIONS
a(14)-a(29) from Alois P. Heinz, Aug 08 2015
STATUS
approved