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

%I A093669 #11 Nov 14 2021 10:15:33
%S A093669 11,14,17,19,20,27,32,34,36,43,46,49,52,64,67,73,82,97,100,142,148,
%T A093669 163,193
%N A093669 Numbers having a unique representation as ab+ac+bc, with 0 < a < b < c.
%C A093669 Are there more terms?
%C A093669 No more terms < 10^6. - _Joerg Arndt_, Oct 01 2017
%D A093669 See A025052.
%e A093669 11 is on the list because 11 = 1*2+1*3+2*3.
%t A093669 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
%o A093669 (Python)
%o A093669 from collections import Counter
%o A093669 def aupto(N):
%o A093669     acount = Counter()
%o A093669     for i in range(1, N-1):
%o A093669         for j in range(i+1, N//i + 1):
%o A093669             p, s = i*j, i+j
%o A093669             for k in range(j+1, (N-p)//s + 1):
%o A093669                 acount.update([p + s*k])
%o A093669     return sorted([k for k in acount if acount[k] == 1])
%o A093669 print(aupto(10**5)) # _Michael S. Branicky_, Nov 14 2021
%Y A093669 Cf. A025052, A025058, A025060.
%Y A093669 Cf. A000926 (numbers not of the form ab+ac+bc, 0<a<b<c).
%K A093669 nonn,more
%O A093669 1,1
%A A093669 _T. D. Noe_, Apr 08 2004