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 A258119 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all perfect partitions of n. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p 123. LINKS Alois P. Heinz, Rows n = 0..500, flattened EXAMPLE 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; ... MAPLE with(numtheory): T:= proc(m) local b, ll, p; if m=0 then return 1 fi; p:= proc(l) ll:=ll, 2^(l-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 MATHEMATICA T = {1}; T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, 2^(l[]-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 *) CROSSREFS 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 KEYWORD nonn,look,tabf AUTHOR Emeric Deutsch, Jun 07 2015 STATUS approved

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Last modified March 20 22:04 EDT 2023. Contains 361391 sequences. (Running on oeis4.)