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A125225
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Numbers n such that n-1 can be represented as a sum of a subset of divisors of n.
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2
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1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252
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OFFSET
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1,2
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COMMENTS
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The definition is related to that for semiperfect numbers (A005835). Every practical number (A005153) belongs to this sequence but not necessarily vice versa; e.g., 70 is in this sequence but not practical. Every number n in this sequence has sigma(n) >= 2n-1 (A103288) but, despite being abundant, 102 is not in this sequence.
Such numbers can be used to construct inheritance puzzles of the type described by Premchand Anne (see link).
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LINKS
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EXAMPLE
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70 is in this sequence because 70-1=69=35+14+10+7+2+1 and all numbers in the sum are divisors of 70.
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MAPLE
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ss:= proc(n, S) local s, Sp; option remember;
if n = 0 then return true
elif S = {} then return false
fi;
s:= max(S);
if s > n then return procname(n, select(`<=`, S, n))
elif s = n then return true
fi;
s:= min(S);
Sp:= subs(s=NULL, S);
if s > n then false
else procname(n-s, Sp) or procname(n, Sp)
fi
end proc:
select(n -> ss(n-1, numtheory:-divisors(n)), [$1..1000]); # Robert Israel, Aug 05 2016
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MATHEMATICA
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okQ[n_] := With[{dd = Divisors[n]}, AnyTrue[Range[Length[dd], 1, -1], AnyTrue[Subsets[dd, {#}], Total[#] == n-1&]&]]; okQ[1] = True;
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PROG
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(PARI) padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b); ); b; }
isok(n) = {if (n == 1, return (1)); d = divisors(n); nbd = #d; for (i = 1, 2^nbd-1, b = padbin(i, nbd); s = sum(j = 1, nbd, d[j]*b[j]); if (s == (n - 1), return (1)); ); return (0); } \\ Michel Marcus, Aug 30 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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