%I A055653
%S A055653 1,2,3,3,5,6,7,5,7,10,11,9,13,14,15,9,17,14,19,15,21,22,23,15,21,26,19,
%T A055653 21,29,30,31,17,33,34,35,21,37,38,39,25,41,42,43,33,35,46,47,27,43,42,
%U A055653 51,39,53,38,55,35,57,58,59,45,61,62,49,33,65,66,67,51,69,70,71,35,73
%N A055653 Sum of phi(d) [A000010] over all unitary divisors d of n (that is, GCD(d,
               n/d) = 1).
%C A055653 Phi-summation over d-s if runs over all divisors is n, so these values 
               are not exceeding n. Compare also other "Phi-summations" like A053570, 
               A053571, or distinct primes dividing n, etc.
%C A055653 a(n) is also the number of solutions of x^(k+1)=x mod n for some k>=1. 
               - S. R. Finch (Steven.Finch(AT)inria.fr), Apr 11 2006
%C A055653 An integer a is called regular (mod n) if there is an integer x such 
               that a^2 x == a (mod n). Then a(n) is also the number of regular 
               integers a (mod n) such that 1 <= a <= n. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), 
               Sep 04 2008]
%C A055653 Equals row sums of triangle A156361 and inverse Mobius transform (A051731) 
               of A114810. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 28 
               2009]
%D A055653 V. S. Joshi, Order-free integers (mod m), Number Theory (Mysore, 1981), 
               Lect. Notes in Math. 938, Springer-Verlag, 1982, pp. 93-100.
%D A055653 J. Morgado, Inteiros regulares m\'odulo n, Gazeta de Matematica (Lisboa), 
               33 (1972), no. 125-128, 1-5. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), 
               Sep 04 2008]
%D A055653 J. Morgado, A property of the Euler phi-function concerning the integers 
               which are regular modulo n, Portugal. Math., 33 (1974), 185-191. 
               [From Laszlo Toth (ltoth(AT)ttk.pte.hu), Sep 04 2008]
%H A055653 T. D. Noe, <a href="b055653.txt">Table of n, a(n) for n=1..1000</a>
%H A055653 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0605019">Idempotents 
               and Nilpotents Modulo n</a> (arXiv:math.NT/0605019)
%H A055653 L. T\'oth, <a href="http://front.math.ucdavis.edu/0710.1936"> Regular 
               integers modulo $n$</a>, Annales Univ. Sci. Budapest., Sect. Comp., 
               29 (2008), 263-275. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), Sep 
               04 2008]
%F A055653 If n = Product p_i^e_i, a(n) = Product (1+p_i^e_i-p_i^(e_i-1)). - Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Apr 19 2001
%e A055653 n=1260 has 36 divisors of which 16 are unitary ones: {1,4,5,7,9,20,28,
               35,36,45,63,140,180,252,315,1260} EulerPhi values of these divisors 
               are: {1,2,4,6,6,8,12,24,12,24,36,48,48,72,144,288} The sum is 735, 
               thus a(1260)=735.
%e A055653 Or, 1260=2^2*3^2*5*7, thus a(1260)=(1+2^2-2)*(1+3^2-3)*(1+5-5^0)*(1+7-7^0)=735.
%p A055653 A055653 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 
               2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]^ifactors(n)[ 2 
               ] [ i ] [ 2 ]-ifactors(n)[ 2 ][ i ] [ 1 ]^(ifactors(n)[ 2 ] [ i ] 
               [ 2 ]-1)): od: RETURN(ans) end:
%Y A055653 Cf. A000010, A053570, A053571, A000188, A006833, A055654.
%Y A055653 A157361, A114810 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 28 
               2009]
%Y A055653 Sequence in context: A052274 A085314 A085310 this_sequence A155918 A097248 
               A097247
%Y A055653 Adjacent sequences: A055650 A055651 A055652 this_sequence A055654 A055655 
               A055656
%K A055653 nonn,easy,nice,mult
%O A055653 1,2
%A A055653 Labos E. (labos(AT)ana.sote.hu), Jun 07 2000