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A258118
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Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all complete partitions of n.
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4
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1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 30, 36, 40, 48, 64, 42, 54, 56, 60, 72, 80, 96, 128, 84, 90, 100, 108, 112, 120, 144, 160, 192, 256, 126, 132, 140, 150, 162, 168, 176, 180, 200, 216, 224, 240, 288, 320, 384, 512, 198, 210, 220, 252, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 576, 640, 768, 1024
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OFFSET
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0,2
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COMMENTS
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A partition of n is complete if every number from 1 to n can be represented as a sum of parts of the partition.
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,1,4] we get 2*2*2*7 = 56. It is in the sequence because the partition [1,1,1,4] is complete.
Except for a(0)=1, there are no odd numbers in the sequence. Indeed, a partition having an odd Heinz number does not have 1 as a part and, consequently, it cannot be complete.
Number of terms in row n is A126796(n). As a matter of fact, so far, the triangle has been constructed by selecting those A126796(n) entries from row n of A215366 which correspond to complete partitions. Last term in row n is 2^n.
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LINKS
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EXAMPLE
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54 = 2*3*3*3 is in the sequence because the partition [1,2,2,2] is complete.
28 = 2*2*7 is not in the sequence because the partition [1,1,4] is not complete.
Triangle T(n,k) begins:
1;
2;
4;
6, 8;
12, 16;
18, 20, 24, 32;
30, 36, 40, 48, 64;
42, 54, 56, 60, 72, 80, 96, 128;
84, 90, 100, 108, 112, 120, 144, 160, 192, 256;
...
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MAPLE
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T:= proc(m) local b, ll, p;
p:= proc(l) ll:=ll, (mul(ithprime(j), j=l)); 1 end:
b:= proc(n, i, l) `if`(i<2, p([l[], 1$n]), `if`(n<2*i-1,
b(n, iquo(n+1, 2), l), b(n, i-1, l)+b(n-i, i, [l[], i])))
end: ll:= NULL; b(m, iquo(m+1, 2), []): sort([ll])[]
end:
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MATHEMATICA
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T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, Product[Prime[j], {j, l}]]; 1); b[n_, i_, l_List] := If[i<2, p[Join[l, Array[1&, n]]], If[n < 2*i-1, b[n, Quotient[n+1, 2], l], b[n, i-1, l] + b[n-i, i, Append[l, i] ]]]; ll = {}; b[m, Quotient[m+1, 2], {}]; Sort[ll]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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