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%I #53 Oct 30 2023 07:29:49
%S 6,3,5,7,22,11,13,27,17,19,46,23,124,58,29,31,250,57,37,55,41,43,94,
%T 47,1264,106,53,87,118,59,61,85,134,67,142,71,73,712,158,79,166,83,
%U 405,115,89,141,406,119,97,202,101,103,214,107,109,145,113,177,418,143
%N Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.
%C a(0) = 6 corresponds to the smallest perfect number.
%C Is n = 144 the first number for which a(n) = 0? - _T. D. Noe_, Mar 28 2013
%C No, a(144) = 95501968. - _Giovanni Resta_, Mar 28 2013
%C We can instead compute k - sigma(k)/2 for increasing k, which is computationally much faster. In this case, we stop computing when all n have been found for a range of numbers. - _T. D. Noe_, Mar 28 2013
%C Also, the first number whose deficiency is 2n. This is the even bisection of A082730. Hence, the first number in the following sequences: A000396, A191363, A125246, A141548, A125247, A101223, A141549, A141550, A125248, A223608, A223607, A223606. - _T. D. Noe_, Mar 29 2013
%C 10^12 < a(654) <= 618970019665683124609613824. - _Donovan Johnson_, Jan 04 2014
%H Donovan Johnson, <a href="/A217769/b217769.txt">Table of n, a(n) for n = 0..653</a> (first 241 terms from T. D. Noe)
%H Nichole Davis, Dominic Klyve and Nicole Kraght, <a href="http://dx.doi.org/10.2140/involve.2013.6.493">On the difference between an integer and the sum of its proper divisors</a>, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
%e a(4)=22, since 22 is the least number such that sigma(22)=36=2*(22-4).
%t Table[Min[Select[Range[2000], DivisorSigma[1, #] == 2*(# - i) &]], {i, 0, 60}]
%t nn = 144; t = Table[0, {nn}]; k = 0; While[k++; Times @@ t == 0, s = (2*k - DivisorSigma[1, k])/2; If[s >= 0 && s < nn && IntegerQ[s] && t[[s + 1]] == 0, t[[s + 1]] = k]]; t (* _T. D. Noe_, Mar 28 2013 *)
%Y Cf. A000203, A000396.
%Y Cf. A087998 (negative n).
%K nonn
%O 0,1
%A _Jayanta Basu_, Mar 28 2013
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