A065387 a(n) = sigma(n) + phi(n).

2, 4, 6, 9, 10, 14, 14, 19, 19, 22, 22, 32, 26, 30, 32, 39, 34, 45, 38, 50, 44, 46, 46, 68, 51, 54, 58, 68, 58, 80, 62, 79, 68, 70, 72, 103, 74, 78, 80, 106, 82, 108, 86, 104, 102, 94, 94, 140, 99, 113, 104, 122, 106, 138, 112, 144, 116, 118, 118, 184, 122, 126, 140

Offset: 1

Author

Labos Elemer, Nov 05 2001

Keywords

• nonn
• easy

Comments

a(n) = 2n for n listed in A008578, the prime numbers at the beginning of the 20th century. When a(n) = a(n + 1), n is probably listed in A066198, numbers n where phi changes as fast as sigma (the only exceptions below 10000 are 2 and 854). - Alonso del Arte, Nov 16 2005

A. Makowski proved that n is prime if and only if a(n) = n * d(n), where d is A000005. - Charles R Greathouse IV, Mar 19 2012

If n is semiprime, a(n) = 2n+1+ceiling(sqrt(n))-floor(sqrt(n)). - Wesley Ivan Hurt, May 05 2015

Atanassov proves that a(n) >= n + A001414(n). - Charles R Greathouse IV, Dec 06 2016

a(n) = 2*n+1 iff n is square of prime (A001248), a(n) = 2*(n+1) iff n is squarefree semiprime (A006881). - Bernard Schott, Feb 09 2020

Examples

			a(10) = 22 because there are 4 coprimes to 10 below 10, the divisors of 10 add up to 18, and 4 + 18 = 22.
		

References

• K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe.)
• A. Makowski, Aufgaben 339, Elemente der Mathematik 15 (1960), pp. 39-40.

Crossrefs

Cf. A000010, A000203, A065388, A015702, A051709, A011774.

See A292768 for partial sums, A051612 for sigma - phi.

Programs

Magma

[DivisorSigma(1,k)+EulerPhi(k):k in [1..65]]; // Marius A. Burtea, Feb 09 2020

Maple

with(numtheory); A065387:=n->phi(n) + sigma(n); seq(A065387(n), n=1..100); # Wesley Ivan Hurt, Apr 08 2014

Mathematica

Table[EulerPhi[n] + DivisorSigma[1,n], {n, 65}] (* Alonso del Arte *)
a[n_] := SeriesCoefficient[Sum[(1+MoebiusMu[k])*x^k/(1-x^k)^2, {k, 1, n}], {x, 0, n}]; Array[a, 63] (* Jean-François Alcover, Sep 29 2017, after Ilya Gutkovskiy *)

PARI

a(n) = sigma(n) + eulerphi(n) \\ Harry J. Smith, Oct 17 2009

SageMath

[sigma(n,1)+euler_phi(n) for n in range(1, 64)] # Stefano Spezia, Jul 20 2025

Formula

a(n) = A000203(n) + A000010(n).

a(n) = A051709(n) + 2n. - N. J. A. Sloane, Jun 12 2004

G.f.: Sum_{k>=1} (mu(k) + 1)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 29 2017