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%I #9 Jan 02 2022 09:50:04
%S 6,12,18,20,24,28,30,36,40,42,45,48,54,56,60,63,66,70,72,78,80,84,88,
%T 90,96,99,100,102,104,105,108,110,112,114,117,120,126,130,132,135,138,
%U 140,144,150,154,156,160,162,165,168,170,174,176,180,182,186,189,192,195
%N Numbers k such that the number of distinct nonzero sums of distinct divisors of k is less than 2^tau(k) - 1 (the largest number of possible distinct sums, tau(k) being the number of divisors of k (A000005)).
%C Equivalently, numbers k for which there exist two distinct subsets of the set of divisors of k having the same sum.
%C 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).
%C 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.
%C Subsequence of A051774 (_Max Alekseyev_).
%H Amiram Eldar, <a href="/A119357/b119357.txt">Table of n, a(n) for n = 1..10000</a>
%e 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 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 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);
%t q[n_] := Module[{d = Divisors[n], x}, Max[CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, Total[d]}], x]] > 1]; Select[Range[200], q] (* _Amiram Eldar_, Jan 02 2022 *)
%Y Cf. A000005, A051774, A119347, A119348, A119425.
%K nonn
%O 1,1
%A _Emeric Deutsch_, May 18 2006
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