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A119348 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct divisors of n.

%I A119348 #8 Aug 06 2024 09:57:00
%S A119348 1,1,2,3,1,3,4,1,2,3,4,5,6,7,1,5,6,1,2,3,4,5,6,7,8,9,10,11,12,1,7,8,1,
%T A119348 2,3,4,5,6,7,8,9,10,11,12,13,14,15,1,3,4,9,10,12,13,1,2,3,5,6,7,8,10,
%U A119348 11,12,13,15,16,17,18,1,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18
%N A119348 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct divisors of n.
%C A119348 Row n contains A119347(n) terms. In row n the first term is 1 and the last term is sigma(n) (=sum of the divisors of n =A000203(n)). If row n contains all numbers from 1 to sigma(n), then n is called a practical number (A005153).
%H A119348 T. D. Noe, <a href="/A119348/b119348.txt">Rows n=1..100, flattened</a>
%e A119348 Row 5 is 1,5,6, the possible sums obtained from the divisors 1 and 5 of 5.
%e A119348 Triangle starts:
%e A119348 1;
%e A119348 1,2,3;
%e A119348 1,3,4;
%e A119348 1,2,3,4,5,6,7;
%e A119348 1,5,6;
%e A119348 1,2,3,4,5,6,7,8,9,10,11,12;
%e A119348 1,7,8;
%e A119348 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
%e A119348 1,3,4,9,10,12,13;
%p A119348 with(numtheory): with(linalg): sums:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): {seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)} end: for n from 1 to 12 do sums(n) od; # yields sequence in triangular form
%t A119348 row[n_] := Union[Total /@ Subsets[Divisors[n]]] // Rest;
%t A119348 Table[row[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Aug 06 2024 *)
%Y A119348 Cf. A000203, A005153, A119347.
%K A119348 nonn,tabf
%O A119348 1,3
%A A119348 _Emeric Deutsch_, May 15 2006