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A254571
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Least multiplier k such that k*n is abundant or perfect (A023196).
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2
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6, 3, 2, 3, 4, 1, 4, 3, 2, 2, 6, 1, 6, 2, 2, 3, 6, 1, 6, 1, 2, 3, 6, 1, 4, 3, 2, 1, 6, 1, 6, 3, 2, 3, 2, 1, 6, 3, 2, 1, 6, 1, 6, 2, 2, 3, 6, 1, 4, 2, 2, 2, 6, 1, 4, 1, 2, 3, 6, 1, 6, 3, 2, 3, 4, 1, 6, 3, 2, 1, 6, 1, 6, 3, 2, 3, 4, 1, 6, 1, 2, 3, 6, 1, 4, 3, 2
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OFFSET
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1,1
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COMMENTS
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See A254572(n)=a(n)*n for the actual non-deficient numbers.
The range is {1,2,3,4,6}. Clearly a(n) <= 6 because 6*n is abundant for any n. No n can have a(n)=5. Suppose otherwise. There exists a prime p smaller than 5 which does not divide n (if not, 6|n and a(n)=1). That prime p (either 2 or 3) will boost the abundancy more than does 5. In particular (sigma(p*n))/(p*n) > (sigma(5*n))/(5*n) >= 2, but then a(n) should be p, a contradiction.
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..10000
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FORMULA
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a(A023196(n)) = 1. - Michel Marcus, Feb 02 2015
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MAPLE
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f:= proc(n) local k; uses numtheory;
for k from 1 to 4 do if sigma(k*n)>=2*k*n then return k fi od:
6
end proc:
map(f, [$1..100]); # Robert Israel, Feb 10 2019
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MATHEMATICA
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a[n_] := Do[If[DivisorSigma[1, k*n] >= 2*k*n, Return[k]], {k, {1, 2, 3, 4, 6}}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 09 2023 *)
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PROG
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(PARI) a(n) = for(k=1, 6, if(sigma(k*n)>=2*k*n, return(k)))
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CROSSREFS
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Cf. A023196, A254572.
Sequence in context: A068996 A068924 A106224 * A360562 A244815 A226579
Adjacent sequences: A254568 A254569 A254570 * A254572 A254573 A254574
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KEYWORD
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nonn,easy
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AUTHOR
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Jeppe Stig Nielsen, Feb 01 2015
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STATUS
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approved
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