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A119028
Numbers having at least 3 unique partitions into exactly 3 parts with the same product.
5
39, 45, 49, 53, 62, 64, 65, 70, 71, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
OFFSET
1,1
COMMENTS
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.
All integers >= 103 are members of this sequence: see second comment in A103277. - Charles Kluepfel and M. F. Hasler, Nov 23 2018
EXAMPLE
49 = 7 + 18 + 24 7*18*24 = 3024
49 = 8 + 14 + 27 8*14*27 = 3024
49 = 9 + 12 + 28 9*12*28 = 3024
or
49 = 9 + 20 + 20 9*20*20 = 3600
49 = 10 + 15 + 24 10*15*24 = 3600
49 = 12 + 12 + 25 12*12*25 = 3600
MATHEMATICA
pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
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]] ]]];
Select[ Range[4, 121], tanya@# >= 3 (*or strictly = ?*) &]
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 &]
CROSSREFS
KEYWORD
nonn
AUTHOR
Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 23 2006, Aug 10 2006
EXTENSIONS
More terms from Robert G. Wilson v, Jul 27 2006
STATUS
approved