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Possible values of sigma(n)-n.
6

%I #22 Oct 26 2023 08:42:53

%S 0,1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,

%U 50,51,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74

%N Possible values of sigma(n)-n.

%C To test whether k>1 is in the sequence, it suffices to check values of n up to (k-1)^2, since sigma(n)-n >= sqrt(n)+1 if n is composite.

%C Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - _N. J. A. Sloane_, Dec 21 2019

%C The lower asymptotic density is at least 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 1 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number absent from this sequence.) - _Charles R Greathouse IV_, Dec 14 2022

%H Alois P. Heinz, <a href="/A078923/b078923.txt">Table of n, a(n) for n = 1..1000</a>

%H Don Coppersmith, <a href="/A001065/a001065.pdf">An answer to the problem of Don Saari</a>, 1987.

%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1973-27.pdf">Über die Zahlen der Form sigma(n)-n und n-phi(n)</a>, Elemente der Math., Vol. 28 (1973), pp. 83-86; <a href="https://www.digizeitschriften.de/en/dms/met/?PPN=PPN378850199_0028&amp;DMDID=dmdlog30">alternative link</a>.

%o (PARI) lista(nn)=for (n=0, nn, if (n==1, kmax=2, kmax=(n-1)^2); for (k=1, kmax, if (sigma(k)-k == n, print1(n, ", "); break););); \\ _Michel Marcus_, Nov 11 2014

%Y Cf. A000203, A001065, A002191, A007369. Complement of A005114.

%K nonn

%O 1,3

%A _Benoit Cloitre_, Dec 15 2002

%E Edited by _Dean Hickerson_, Dec 19 2002

%E Offset fixed by _Michel Marcus_, Dec 19 2014