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

1,1

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

Links

Table of n, a(n) for n=1..60.

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

Cf. A069905, A103277, A317578.

Sequence in context: A309696 A061756 A236673 * A252712 A252711 A046512

Adjacent sequences: A119025 A119026 A119027 * A119029 A119030 A119031

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