%I
%S 39,45,49,53,62,64,65,70,71,74,75,76,77,78,79,81,82,83,84,85,87,88,89,
%T 90,91,92,93,94,95,97,98,99,100,101,103,104,105,106,107,108,109,110,
%U 111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128
%N Numbers having at least 3 unique partitions into exactly 3 parts with the same product.
%C That is, numbers j such that there exist positive integers a1 <= a2 <= a3, b1 <= b2 <= b3, c1 <= c2 <= c3 (unique as triples) with j = a1 + a2 + a3 = b1 + b2 + b3 = c1 + c2 + c3 and a1*a2*a3 = b1*b2*b3 = c1*c2*c3. The answer to a question raised by _Tanya Khovanova_, Jul 23 2006.
%C All integers >= 103 are members of this sequence: see second comment in A103277.  _Charles Kluepfel_ and _M. F. Hasler_, Nov 23 2018
%e 49 = 7 + 18 + 24 7*18*24 = 3024
%e 49 = 8 + 14 + 27 8*14*27 = 3024
%e 49 = 9 + 12 + 28 9*12*28 = 3024
%e or
%e 49 = 9 + 20 + 20 9*20*20 = 3600
%e 49 = 10 + 15 + 24 10*15*24 = 3600
%e 49 = 12 + 12 + 25 12*12*25 = 3600
%t pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
%t tanya[n_] := Max[Length /@ Split[Sort[pdt /@ Union[ Partition[Last /@ Flatten[ FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c}, Integers,(* failsafe *) PartitionsP@n]], 3]] ]]];
%t Select[ Range[4, 121], tanya@# >= 3 (*or strictly = ?*) &]
%t Select[Range[3, 121], Max[Length /@ Split[Sort[Times @@@ Partition[Last /@ Flatten[FindInstance[a + b + c == # && a >= b >= c > 0, {a, b, c}, Integers,(* cf A069905 *) Round[ #^2/12]]], 3]]]] >= 3 &]
%Y Cf. A069905, A103277, A317578.
%K nonn
%O 1,1
%A Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 23 2006, Aug 10 2006
%E More terms from _Robert G. Wilson v_, Jul 27 2006
