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A355641
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Numbers k that can be written as the sum of 5 divisors of k (not necessarily distinct).
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10
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5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 78, 80, 81, 84, 85, 88, 90, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 120, 125, 126, 128, 130, 132, 135, 136, 138
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OFFSET
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1,1
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COMMENTS
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Numbers that are divisible by at least one of 5, 6, 8, 9, 14 and 21. For proof see link. - Robert Israel, Sep 01 2022
The asymptotic density of this sequence is 17/35. - Amiram Eldar, Aug 08 2023
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LINKS
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EXAMPLE
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9 is in the sequence since 9 = 3+3+1+1+1, where each summand divides 9.
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MAPLE
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F:= proc(x, S, j) option remember;
local s, k;
if j = 0 then return(x = 0) fi;
if S = [] or x > j*S[-1] or x < j*S[1] then return false fi;
s:= S[-1];
for k from 0 to min(j, floor(x/s)) do
if procname(x-k*s, S[1..-2], j-k) then return true fi
od;
false
end proc:
select(t -> F(t, sort(convert(numtheory:-divisors(t), list)), 5), [$1..200]); # Robert Israel, Aug 31 2022
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MATHEMATICA
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q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[140], q[#, 5] &] (* Amiram Eldar, Aug 19 2022 *)
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PROG
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(PARI) isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [5, 5]); \\ Michel Marcus, Aug 19 2022
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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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