A101902 Numbers n that are not of the form ab+bc+cd+de+ea with 1<=a<=b<=c<=d<=e.

1, 2, 3, 4, 6, 8, 12, 14, 18, 30, 38, 42, 62

Offset: 1

Author

T. D. Noe, Dec 20 2004

Keywords

• nonn
• fini
• full

Comments

Conjecture that this sequence is complete. Note that, except for n=1 and n=2, n-1 is prime. The case of the 3-term binary form is treated in A025052. For the 4-term 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

Crossrefs

Cf. A025052 (n not of form ab + bc + ca).

Programs

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]]

Extensions

Definition corrected by Ivan Neretin, Dec 14 2016