A348824 Numbers in array A327259 that do not have a unique decomposition into numbers of A327261.

32, 48, 72, 96, 112, 126, 128, 144, 160, 168, 176, 192, 198, 221, 224, 240, 252, 256, 264, 288, 294, 304, 336, 342, 347, 352, 360, 368, 384, 392, 396, 414, 416, 432, 448, 456, 462, 480, 496, 504, 512, 528, 544, 545, 552, 558, 560, 576, 588, 599

Offset

1,1

Comments

While array A327259 has many properties of the multiplication table, one way the numbers that sieve out of the array fail to be prime numbers is that unique factorization does not hold. Some numbers have two or more decompositions.

For i >= 2, A327259(i, a(n)) is in the sequence.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1256 (all terms m <= 10000)

David Lovler, Decomposition of a(n) into A327261(k)

Example

48 is in the sequence because 48 = A327259(2,12) = A327259(4,6) and 2, 4, 6 and 12 are in A327261.
72 is in the sequence because 72 = A327259(2,2,5) = A327259(6,6) and 2, 5 and 6 are in A327261. A327259(2,2,5) is well-defined because A327259(n,k) is associative.
221 is in the sequence because 221 = A327259(5,25) = A327259(11,11) and 5, 11 and 25 are in A327261.
462 is in the sequence because 462 = A327259(6,39) = A327259(11,22) = A327259(14,17) and 6, 11, 14, 17, 22 and 39 are in A327261.
The first six terms and their decompositions:
1 32 = A327259(2,2,2) = A327259(4,4)
2 48 = A327259(2,12) = A327259(4,6)
3 72 = A327259(2,2,5) = A327259(6,6)
4 96 = A327259(2,2,6) = A327259(4,12)
5 112 = A327259(2,28) = A327259(4,14)
6 126 = A327259(5,14) = A327259(6,11)
More in a-file.

Mathematica

T[n_, k_]:=2n*k-If[Mod[n, 2]==1, If[Mod[k, 2]==1, n+k-1, k], If[Mod[k, 2]==1, n, 0]]; F[d_]:=If[(q=Union[Sort/@(Position[Table[T[n, k], {n, 2, Ceiling[d/3]}, {k, 2, Ceiling[d/3]}], d]+1)])=={}, {{d}}, q]; FC[x_]:=FixedPoint[Union[Sort/@Flatten[Flatten/@Tuples[#]&/@((F/@#&/@#)&[#]), 1]]&, F[x]]; list={}; Do[If[Length@FC@i>1, AppendTo[list, i]], {i, 300}]; list (* Giorgos Kalogeropoulos, Nov 05 2021 *)

Crossrefs

Cf. A327259, A327261, A340747.

Sequence in context: A036329 A014614 A046371 * A175162 A114437 A039379

Adjacent sequences: A348821 A348822 A348823 * A348825 A348826 A348827

Keyword

nonn

Author

David Lovler, Oct 31 2021

Extensions

Name amended by David Lovler, Jan 26 2022

Status

approved