%I A119357
%S A119357 6,12,18,20,24,28,30,36,40,42,45,48,54,56,60,63,66,70,72,78,80,84,88,90,
%T A119357 96,99,100,102,104,105,108,110,112,114,117,120,126,130,132,135,138,140,
%U A119357 144,150,154,156,160,162,165,168,170,174,176,180,182,186,189,192,195
%N A119357 Values of n such that the number of distinct nonzero sums of distinct 
               divisors of n is less than 2^tau(n) - 1 (clearly, the latter is the 
               largest number of possible distinct sums, tau(n) being the number 
               of divisors of n (A000005)). Equivalently: values of n for which 
               there exist two distinct subsets of the set of divisors of n, having 
               the same sum.
%C A119357 The sequence is closed with respect to multiplication by positive integers 
               (i.e. any multiple of any term in the sequence is in the sequence). 
               The primitive entries of the sequence, i.e. those that are not multiples 
               of other terms of the sequence, are given in A119425 (the first five 
               are 6,20,28,45 and 63). The number of distinct sums of distinct divisors 
               of n are given in A119347 and the actual sums are given in row n 
               of the triangle A119348. Subsequence of A051774 (Max Alekseyev)
%e A119357 6 is in the sequence because from the divisors of 6, namely 1,2,3,6, 
               we can form by addition 12 sums (1,2,3,...,12) and 12 < 2^tau(6)-1=2^4-1=15.
%e A119357 Sequence contains, for example, all multiples of 6 (1+2=3), all multiples 
               of 20 (1+4=5), all multiples of 28 (1+2+4=7), all multiples of 63 
               (1+9=3+7).
%p A119357 with(numtheory): with(linalg): s:=proc(n) local dl,t:dl:=convert(divisors(n),
               list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),
               i=1..2^t-1)}) end: a:=proc(n) if s(n)<2^tau(n)-1 then n else fi end: 
               seq(a(n),n=1..230);
%Y A119357 Cf. A000005, A051774, A119347, A119348, A119425.
%Y A119357 Sequence in context: A049094 A105289 A051774 this_sequence A097216 A023196 
               A005835
%Y A119357 Adjacent sequences: A119354 A119355 A119356 this_sequence A119358 A119359 
               A119360
%K A119357 nonn
%O A119357 1,1
%A A119357 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 18 2006