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A258119
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Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all perfect partitions of n.
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5
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1, 2, 4, 6, 8, 16, 18, 20, 32, 64, 42, 54, 56, 128, 100, 256, 162, 176, 512, 1024, 234, 260, 294, 392, 416, 486, 500, 2048, 4096, 1088, 1458, 8192, 1936, 2500, 16384, 798, 1026, 1064, 2058, 2432, 2744, 4374, 32768, 65536, 2300, 3042, 3380, 5408, 5888, 12500, 13122, 131072
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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 perfect if it contains just one partition of every number less than n when repeated parts are regarded as indistinguishable.
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 perfect.
Number of terms in row n is A002033(n). As a matter of fact, so far the triangle has been constructed by selecting those A002033(n) entries from row n of A215366 which correspond to perfect partitions. Last term in row n is 2^n.
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p 123.
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LINKS
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Alois P. Heinz, Rows n = 0..500, flattened
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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 perfect.
24 = 2*2*2*3 is not in the sequence because the partition [1,1,1,2] is not perfect (1+1+1=1+2; it is complete).
Triangle T(n,k) begins:
1;
2;
4;
6, 8;
16;
18, 20, 32;
64;
42, 54, 56, 128;
...
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MAPLE
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with(numtheory):
T:= proc(m) local b, ll, p;
if m=0 then return 1 fi;
p:= proc(l) ll:=ll, 2^(l[1]-1)*mul(ithprime(
mul(l[j], j=1..i-1))^(l[i]-1), i=2..nops(l)) end:
b:= proc(n, l) `if`(n=1, p(l), seq(b(n/d, [l[], d])
, d=divisors(n) minus{1})) end:
ll:= NULL; b(m+1, []): sort([ll])[]
end:
seq(T(n), n=0..20); # Alois P. Heinz, Jun 08 2015
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MATHEMATICA
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T[0] = {1}; T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, 2^(l[[1]]-1)*Product[Prime[Product[l[[j]], {j, 1, i-1}]]^(l[[i]]-1), {i, 2, Length[l]}]]; 1); b[n_, l_List] := If[n == 1, p[l], Table[b[n/d, Append[l, d]], {d, Divisors[n] // Rest}]]; ll = {}; b[m+1, {}]; Sort[ll] ]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)
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CROSSREFS
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Cf. A000079, A215366, A002033, A258118.
Column k=1 gives A259939.
Row sums give A360713.
Sequence in context: A283423 A073935 A325764 * A073696 A058602 A133808
Adjacent sequences: A258116 A258117 A258118 * A258120 A258121 A258122
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KEYWORD
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nonn,look,tabf
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AUTHOR
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Emeric Deutsch, Jun 07 2015
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STATUS
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approved
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