A119028 Numbers having at least 3 unique partitions into exactly 3 parts with the same product.

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

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 23 2006, Aug 10 2006

Keywords

• nonn

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

Examples

			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
		

Crossrefs

Cf. A069905, A103277, A317578.

Programs

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 &]

Extensions

More terms from Robert G. Wilson v, Jul 27 2006