A078923 Possible values of sigma(n)-n.

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, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset

1,3

Comments

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.

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Don Coppersmith, An answer to the problem of Don Saari, 1987.

Paul Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math., Vol. 28 (1973), pp. 83-86; alternative link.

Prog

(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

Crossrefs

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

Sequence in context: A317093 A181590 A243293 * A165240 A350398 A092460

Adjacent sequences: A078920 A078921 A078922 * A078924 A078925 A078926

Keyword

nonn

Author

Benoit Cloitre, Dec 15 2002

Extensions

Edited by Dean Hickerson, Dec 19 2002

Offset fixed by Michel Marcus, Dec 19 2014

Status

approved