This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A078923 #30 Aug 11 2025 08:38:17 %S A078923 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 A078923 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, %U A078923 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 A078923 Possible values of sigma(n)-n. %C A078923 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 A078923 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 A078923 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 A078923 Amiram Eldar, <a href="/A078923/b078923.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz) %H A078923 Don Coppersmith, <a href="/A001065/a001065.pdf">An answer to the problem of Don Saari</a>, 1987. %H A078923 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://gdz.sub.uni-goettingen.de/id/PPN378850199_0028">alternative link</a>. %o A078923 (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 A078923 Cf. A000203, A001065, A002191, A007369. Complement of A005114. %K A078923 nonn %O A078923 1,3 %A A078923 _Benoit Cloitre_, Dec 15 2002 %E A078923 Edited by _Dean Hickerson_, Dec 19 2002 %E A078923 Offset fixed by _Michel Marcus_, Dec 19 2014