A119357 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)).
6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 144, 150, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 186, 189, 192, 195
Offset: 1
Keywords
Examples
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. 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).
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
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); -
Mathematica
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 *)
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