A022904 Number of solutions to c(1)*prime(4) + ... + c(n)*prime(n+3) = 1, where c(i) = +-1 for i>1, c(1) = 1.

0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 6, 0, 32, 0, 110, 0, 252, 0, 1139, 0, 3127, 0, 12743, 0, 39767, 0, 156376, 0, 517381, 0, 1870169, 0, 6786580, 0, 25420402, 0, 90815872, 0, 334621081, 0, 1235976769, 0, 4597232973, 0, 17047065235, 0, 63450750049, 0, 238163814619, 0

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			a(7) counts these 3 solutions: {7, -11, 13, 17, -19, 23, -29}, {7, 11, -13, -17, 19, 23, -29}, {7, 11, 13, -17, -19, -23, 29}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A022903, A022920.

Programs

Maple

A022904 := proc(n)
    local a,b,cs,cslen ;
    a := 0 ;
    for b from 0 to 2^(n-1)-1 do
        cs := convert(b,base,2) ;
        cslen := nops(cs) ;
        if cslen < n-1 then
            cs := [op(cs),seq(0,i=1..n-1-cslen)] ;
        end if;
        if ithprime(4)+add( (-1+2*op(i-4,cs)) *ithprime(i),i=5..n+3) = 1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 06 2015

Mathematica

{f, s} = {4, 1}; Table[t = Map[Prime[# + f - 1] &, Range[2, z]]; Count[Map[Apply[Plus, #] &, Map[t # &, Tuples[{-1, 1}, Length[t]]]], s - Prime[f]], {z, 22}]
(* A022904, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
n = 7; t = Map[Prime[# + f - 1] &, Range[n]]; Map[#[[2]] &, Select[Map[{Apply[Plus, #], #} &, Map[t # &, Map[Prepend[#, 1] &, Tuples[{-1, 1}, Length[t] - 1]]]], #[[1]] == s &]]  (* the 3 solutions of using n=7 primes; Peter J. C. Moses, Oct 01 2013 *)

Formula

a(n) = [x^6] Product_{k=5..n+3} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 30 2024

Extensions

Corrected and extended by Clark Kimberling, Oct 01 2013

a(23)-a(50) from Alois P. Heinz, Aug 06 2015