A281626 a(n) = (sum of trivial divisors of n) - (sum of nontrivial divisors of n).

1, 3, 4, 3, 6, 2, 8, 3, 7, 4, 12, -2, 14, 6, 8, 3, 18, -1, 20, 0, 12, 10, 24, -10, 21, 12, 16, 2, 30, -10, 32, 3, 20, 16, 24, -17, 38, 18, 24, -8, 42, -10, 44, 6, 14, 22, 48, -26, 43, 9, 32, 8, 54, -10, 40, -6, 36, 28, 60, -46, 62, 30, 24, 3, 48, -10, 68, 12

Offset: 1

Author

Jaroslav Krizek, Feb 11 2017

Keywords

• sign

Comments

Trivial divisors of n are numbers 1 and n.

a(n) = 0 for numbers in A088831 (numbers n whose abundance is 2).

a(n) <= 0 for numbers in A005101 (abundant numbers).

a(n) > 0 for numbers in A263837 (non-abundant numbers).

Examples

			a(20) = (20+1) - (2+4+5+10) = 0.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A000396, A005101, A006881, A048050, A088831, A263837.

Programs

Magma

[1] cat [2*(n+1) - SumOfDivisors(n): n in [2..100]];

Mathematica

Table[If[n == 1, 1, 2 (n + 1) - DivisorSigma[1, n]], {n, 68}] (* Michael De Vlieger, Feb 11 2017 *)

Formula

a(1) = 1; for n>1, a(n) = (n+1) - (sigma(n) - n - 1) = 2*(n+1) - sigma(n) = n + 1 - A048050(n).

a(A000396(n)) = 2 for n >= 1.

a(A000079(n)) = 3 for n >= 1.

a(A006881(n)) = phi(n).

a(p) = p + 1 for p prime.