A291784 a(n) = (psi(n) + phi(n))/2.

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 14, 13, 15, 16, 16, 17, 21, 19, 22, 22, 23, 23, 28, 25, 27, 27, 30, 29, 40, 31, 32, 34, 35, 36, 42, 37, 39, 40, 44, 41, 54, 43, 46, 48, 47, 47, 56, 49, 55, 52, 54, 53, 63, 56, 60, 58

Offset: 1

Author

N. J. A. Sloane, Sep 02 2017

Keywords

• nonn
• easy

Comments

This is (A001615 + A000010)/2. It is easy to see that this is always an integer.

If n is a power of a prime (including 1 and primes), then a(n) = n, and in any other case a(n) > n. - M. F. Hasler, Sep 09 2017

If n is in A006881, then a(n)=n+1. - Robert Israel, Feb 10 2019

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41 (page 96 of 2nd ed., pages 147ff of 3rd ed.).

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..10000
• Marcin Mazur and Bogdan V. Petrenko, Generalizations of Arnold's version of Euler's theorem for matrices, Japanese Journal of Mathematics, 5:183-189, 2010.
• N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Crossrefs

Cf. A000010, A001615, A006881, A291785, A291786, A291787, A291788.

Programs

Maple

f:= proc(n) local P, p;
  P:= numtheory:-factorset(n);
  n*(mul((p-1)/p, p=P) + mul((p+1)/p, p=P))/2
end proc:
map(f, [$1..100]); # Robert Israel, Feb 10 2019

Mathematica

psi[n_] := If[n == 1, 1, n*Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := (psi[n] + EulerPhi[n])/2;
Array[a, 100] (* Jean-François Alcover, Feb 25 2019 *)

PARI

A291784(n)=(eulerphi(n)+n*sumdivmult(n,d,issquarefree(d)/d))\2 \\ M. F. Hasler, Sep 03 2017

Formula

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 21/(4*Pi^2) = 0.531936... . - Amiram Eldar, Dec 05 2023