|
|
A261063
|
|
Number of solutions to c(1)*prime(3) + ... + c(2n-1)*prime(2n+1) = -1, where c(i) = +-1 for i > 1, c(1) = 1.
|
|
19
|
|
|
0, 0, 0, 1, 6, 8, 40, 67, 373, 1232, 3330, 13656, 47111, 164957, 582042, 1967152, 7129046, 26655235, 94956602, 353789267, 1300061367, 4765080122, 17726643505, 66038899483, 245431428625, 919911458949, 3457983108462, 12974054097333, 49016641868213, 185510228030858
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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):
|
|
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|