
COMMENTS

Conjecture that this sequence is complete. Note that, except for n=1 and n=2, n1 is prime. The case of the 3term binary form is treated in A025052. For the 4term case, ab+bc+cd+da, no prime is representable because the 4 terms factor as (a+c)(b+d).
The sequence is indeed complete. Each sufficiently great number can be represented in one of the following ways:
n = 7k: {1, 2, 5, 6, k  6}
n = 7k + 1: {1, 1, 1, 6, k  1}
n = 7k + 2: {1, 3, 3, 6, k  4}
n = 7k + 3: {2, 3, 4, 5, k  5}
n = 7k + 4: {1, 2, 2, 6, k  2}
n = 7k + 5: {1, 2, 3, 6, k  3}
n = 7k + 6: {1, 2, 4, 6, k  4}
Smaller numbers can be checked individually.  Ivan Neretin, Dec 14 2016


MATHEMATICA

nn=100; cnt5=Table[0, {nn}]; Do[n=a*b+b*c+c*d+d*e+e*a; If[n<=nn, cnt5[[n]]++ ], {a, nn}, {b, a, nn}, {c, b, nn}, {d, c, nn}, {e, d, nn}]; Flatten[Position[cnt5, 0]]
