A291784 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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)

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

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 *)