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%I #32 Dec 04 2023 01:38:12
%S -1,0,-1,1,-3,4,-5,3,-2,4,-9,12,-11,4,1,7,-15,15,-17,14,-1,4,-21,28,
%T -14,4,-5,16,-27,34,-29,15,-5,4,-11,43,-35,4,-7,34,-39,42,-41,20,9,4,
%U -45,60,-34,23,-11,22,-51,48,-23,40,-13,4,-57,92,-59
%N a(n) = sigma(n) - phi(n) - n.
%C While terms with even indices are never negative, this is the case for most terms with odd indices; exceptions are listed in A229978.
%H Douglas E. Iannucci, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Iannucci/ian5.html">On the Equation sigma(n) = n + phi(n)</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.
%F a(n) = 0 <=> n = 2 (conjectured).
%F a(2n) > 0 for all n > 1.
%F a(2n+1) > 0 <=> n in A229978.
%F a(n) = A051612(n) - n = A000203(n) - A000010(n) - n.
%F a(p) = 2 - p for p prime. - _Alonso del Arte_, Oct 05 2013
%F Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 - 6/Pi^2 - 1 = 0.0370069... . - _Amiram Eldar_, Dec 04 2023
%t Table[DivisorSigma[1, n] - EulerPhi[n] - n, {n, 75}] (* _Alonso del Arte_, Oct 05 2013 *)
%o (PARI) A228947(n)=sigma(n)-eulerphi(n)-n
%Y Cf. A000010, A000203, A051612, A229978.
%K sign,easy
%O 1,5
%A _M. F. Hasler_, Oct 05 2013