A228947 a(n) = sigma(n) - phi(n) - n.

-1, 0, -1, 1, -3, 4, -5, 3, -2, 4, -9, 12, -11, 4, 1, 7, -15, 15, -17, 14, -1, 4, -21, 28, -14, 4, -5, 16, -27, 34, -29, 15, -5, 4, -11, 43, -35, 4, -7, 34, -39, 42, -41, 20, 9, 4, -45, 60, -34, 23, -11, 22, -51, 48, -23, 40, -13, 4, -57, 92, -59

Offset: 1

Author

M. F. Hasler, Oct 05 2013

Keywords

• sign
• easy

Comments

While terms with even indices are never negative, this is the case for most terms with odd indices; exceptions are listed in A229978.

Links

• Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Crossrefs

Cf. A000010, A000203, A051612, A229978.

Programs

Mathematica

Table[DivisorSigma[1, n] - EulerPhi[n] - n, {n, 75}] (* Alonso del Arte, Oct 05 2013 *)

PARI

A228947(n)=sigma(n)-eulerphi(n)-n

Formula

a(n) = 0 <=> n = 2 (conjectured).

a(2n) > 0 for all n > 1.

a(2n+1) > 0 <=> n in A229978.

a(n) = A051612(n) - n = A000203(n) - A000010(n) - n.

a(p) = 2 - p for p prime. - Alonso del Arte, Oct 05 2013

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 - 6/Pi^2 - 1 = 0.0370069... . - Amiram Eldar, Dec 04 2023