A157449 Difference between n and the sum of its divisors except 1 and itself.

2, 3, 2, 5, 1, 7, 2, 6, 3, 11, -3, 13, 5, 7, 2, 17, -2, 19, -1, 11, 9, 23, -11, 20, 11, 15, 1, 29, -11, 31, 2, 19, 15, 23, -18, 37, 17, 23, -9, 41, -11, 43, 5, 13, 21, 47, -27, 42, 8, 31, 7, 53, -11, 39, -7, 35, 27, 59, -47, 61, 29, 23, 2

Offset: 2

Author

Ferruccio Guidi (fguidi(AT)cs.unibo.it), Mar 01 2009

Keywords

• sign
• easy

Comments

a(n) = n - k where k is the sum of the divisors of n excluding 1 and n itself. The initial value for n is 2.

Evidently a(n) = n iff n is prime (A000040). Moreover a(n) = 1 iff n is perfect (A000396).

A value of 0 indicates a quasiperfect number, although no such number is known. - Felix Fröhlich, Jul 14 2014

a(n) is negative iff n is abundant (A005101). - Christian N. K. Anderson, May 02 2023

Examples

			The divisors of 10 are 1, 2, 5 and 10, so a(10) = 10 - (2 + 5) = 3.
		

Links

• Ferruccio Guidi, Table of n, a(n) for n=2..100001
• Wikipedia, Quasiperfect number

Crossrefs

Cf. A000040, A000203, A000396, A005101.

Programs

Mathematica

Table[2n+1-DivisorSigma[1,n],{n,70}] (* Harvey P. Dale, Jul 22 2013 *)

PARI

for(n=2, 1e2, a=2*n+1; b=sigma(n); print1(a-b, ", ")) \\ Felix Fröhlich, Jul 14 2014

Formula

a(n) = (2*n+1)-A000203(n). - Felix Fröhlich, Jul 14 2014