A235796 2*n - 1 - sigma(n).

0, 0, 1, 0, 3, -1, 5, 0, 4, 1, 9, -5, 11, 3, 5, 0, 15, -4, 17, -3, 9, 7, 21, -13, 18, 9, 13, -1, 27, -13, 29, 0, 17, 13, 21, -20, 35, 15, 21, -11, 39, -13, 41, 3, 11, 19, 45, -29, 40, 6, 29, 5, 51, -13, 37, -9, 33, 25, 57, -49, 59, 27, 21, 0, 45, -13, 65, 9, 41

Offset: 1

Author

Omar E. Pol, Jan 25 2014

Keywords

• sign

Comments

Partial sums give A004125.

Also 0 together with A120444.

It appears that a(n) = 0 iff n is a power of 2.

Numbers n with a(n) = 0 are called "almost perfect", "least deficient" or "slightly defective" numbers. See A000079. - Robert Israel, Jul 22 2014

a(n) = n - 2 iff n is prime.

a(n) = -1 iff n is a perfect number.

Also the alternating row sums of A239446. - Omar E. Pol, Jul 21 2014

Examples

			.     The positive     The sum of
n     odd numbers     divisors of n.      a(n)
1          1                1               0
2          3                3               0
3          5                4               1
4          7                7               0
5          9                6               3
6         11               12              -1
7         13                8               5
8         15               15               0
9         17               13               4
10        19               18               1
...
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A000203, A000396, A004125, A005408, A120444, A196020, A236104.

Programs

Magma

[2*n-1-SumOfDivisors(n): n in [1..100]]; // Vincenzo Librandi, Feb 25 2014

Mathematica

Table[2n-1-DivisorSigma[1,n],{n,70}] (* Harvey P. Dale, Jul 11 2014 *)

PARI

vector(100, n, (2*n-1)-sigma(n)) \\ Colin Barker, Jan 27 2014

Formula

a(n) = A005408(n-1) - A000203(n).

a(n) = -1 - A033880(n). - Michel Marcus, Jan 27 2014

a(n) = n - 1 - A001065(n). - Omar E. Pol, Jan 29 2014

a(n) = A033879(n) - 1. - Omar E. Pol, Jan 30 2014

a(n) = 2*n - 2 - A039653(n). - Omar E. Pol, Jan 31 2014

a(n) = (-1)*A237588(n). - Omar E. Pol, Feb 23 2014

a(n) = 2*n - A088580(n). - Omar E. Pol, Mar 23 2014