A093669 Numbers having a unique representation as ab+ac+bc, with 0 < a < b < c.

11, 14, 17, 19, 20, 27, 32, 34, 36, 43, 46, 49, 52, 64, 67, 73, 82, 97, 100, 142, 148, 163, 193

Offset: 1

Author

T. D. Noe, Apr 08 2004

Keywords

• nonn
• more

Comments

Are there more terms?

No more terms < 10^6. - Joerg Arndt, Oct 01 2017

Examples

			11 is on the list because 11 = 1*2+1*3+2*3.
		

References

• See A025052.

Crossrefs

Cf. A025052, A025058, A025060.

Cf. A000926 (numbers not of the form ab+ac+bc, 0

Programs

Mathematica

oneSol={}; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>1, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt==1, AppendTo[oneSol, n]], {n, 10000}]; oneSol

Python

from collections import Counter
def aupto(N):
    acount = Counter()
    for i in range(1, N-1):
        for j in range(i+1, N//i + 1):
            p, s = i*j, i+j
            for k in range(j+1, (N-p)//s + 1):
                acount.update([p + s*k])
    return sorted([k for k in acount if acount[k] == 1])
print(aupto(10**5)) # Michael S. Branicky, Nov 14 2021

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