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A119028 Numbers having at least 3 unique partitions into exactly 3 parts with the same product. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 25 06:59 EDT 2021. Contains 345452 sequences. (Running on oeis4.)