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%I #33 Sep 08 2022 08:46:06
%S 0,0,-1,0,-3,1,-5,0,-4,-1,-9,5,-11,-3,-5,0,-15,4,-17,3,-9,-7,-21,13,
%T -18,-9,-13,1,-27,13,-29,0,-17,-13,-21,20,-35,-15,-21,11,-39,13,-41,
%U -3,-11,-19,-45,29,-40,-6,-29,-5,-51,13,-37,9,-33,-25,-57,49,-59,-27,-21,0
%N Sigma(n) - 2n + 1.
%C Also we can write Sigma(n) - (2n - 1).
%C a(n) = 2 - n iff n is prime.
%C a(n) = 1 iff n is a perfect number.
%C Conjecture: a(n) = 0 iff n is a power of 2.
%C The problem is not new. In fact, the following comments appeared on page 74 of Guy's book: "If Sigma(n) = 2*n - 1, n has been called almost perfect. Powers of 2 are almost perfect; it is not known if any other numbers are.". - _Zhi-Wei Sun_, Feb 23 2014
%D R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004.
%F a(n) = A000203(n) - A005408(n-1) = 1 - n + A001065(n) = 1 - A033879(n) = 1 + A033880(n) = (-1)*A235796(n).
%F a(n) = A088580(n) - 2*n. - _Omar E. Pol_, Mar 23 2014
%e -----------------------------------------------
%e . The sum of The positive
%e n divisors of n odd numbers a(n)
%e -----------------------------------------------
%e 1 1 1 0
%e 2 3 3 0
%e 3 4 5 -1
%e 4 7 7 0
%e 5 6 9 -3
%e 6 12 11 1
%e 7 8 13 -5
%e 8 15 15 0
%e 9 13 17 -4
%e 10 18 19 -1
%e ...
%t Table[DivisorSigma[1,n]-2n+1,{n,70}] (* _Harvey P. Dale_, Nov 15 2014 *)
%o (PARI) vector(100, n, sigma(n)-2*n+1) \\ _Colin Barker_, Feb 21 2014
%o (Magma) [1-2*n+SumOfDivisors(n): n in [1..100]]; // _Vincenzo Librandi_, Feb 25 2014
%Y Cf. A000079, A000203, A000396, A001065, A004125, A005408, A033879, A033880, A039653, A120444, A196020, A235796, A236104.
%K sign
%O 1,5
%A _Omar E. Pol_, Feb 20 2014
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