cp's OEIS Frontend

A258118 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all complete partitions of n.

%I A258118 #34 Jan 05 2025 19:51:40
%S A258118 1,2,4,6,8,12,16,18,20,24,32,30,36,40,48,64,42,54,56,60,72,80,96,128,
%T A258118 84,90,100,108,112,120,144,160,192,256,126,132,140,150,162,168,176,
%U A258118 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
%N A258118 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all complete partitions of n.
%C A258118 A partition of n is complete if every number from 1 to n can be represented as a sum of parts of the partition.
%C A258118 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.
%C A258118 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.
%C A258118 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.
%H A258118 Alois P. Heinz, <a href="/A258118/b258118.txt">Rows n = 0..30, flattened</a>
%H A258118 SeungKyung Park, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-4/park.pdf">Complete Partitions</a>, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.
%e A258118 54 = 2*3*3*3 is in the sequence because the partition [1,2,2,2] is complete.
%e A258118 28 = 2*2*7 is not in the sequence because the partition [1,1,4] is not complete.
%e A258118 Triangle T(n,k) begins:
%e A258118    1;
%e A258118    2;
%e A258118    4;
%e A258118    6,  8;
%e A258118   12, 16;
%e A258118   18, 20,  24,  32;
%e A258118   30, 36,  40,  48,  64;
%e A258118   42, 54,  56,  60,  72,  80,  96, 128;
%e A258118   84, 90, 100, 108, 112, 120, 144, 160, 192, 256;
%e A258118   ...
%p A258118 T:= proc(m) local b, ll, p;
%p A258118       p:= proc(l) ll:=ll, (mul(ithprime(j), j=l)); 1 end:
%p A258118       b:= proc(n, i, l) `if`(i<2, p([l[], 1$n]), `if`(n<2*i-1,
%p A258118       b(n, iquo(n+1, 2), l), b(n, i-1, l)+b(n-i, i, [l[], i])))
%p A258118       end: ll:= NULL; b(m, iquo(m+1, 2), []): sort([ll])[]
%p A258118     end:
%p A258118 seq(T(n), n=0..12);  # _Alois P. Heinz_, Jun 07 2015
%t A258118 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_ *)
%Y A258118 Cf. A000079, A215366, A126796, A258119.
%Y A258118 Column k=1 gives A259941.
%Y A258118 Row sums give A360791.
%K A258118 nonn,look,tabf
%O A258118 0,2
%A A258118 _Emeric Deutsch_, Jun 07 2015