A055653 Sum of phi(d) [A000010] over all unitary divisors d of n (that is, gcd(d,n/d) = 1).

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, 21, 29, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 46, 47, 27, 43, 42, 51, 39, 53, 38, 55, 35, 57, 58, 59, 45, 61, 62, 49, 33, 65, 66, 67, 51, 69, 70, 71, 35, 73

Offset

1,2

Comments

Phi-summation over d-s if runs over all divisors is n, so these values do not exceed n. Compare also other "Phi-summations" like A053570, A053571, or distinct primes dividing n, etc.

a(n) is also the number of solutions of x^(k+1)=x mod n for some k>=1. - Steven Finch, Apr 11 2006

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. - Laszlo Toth, Sep 04 2008

Equals row sums of triangle A157361 and inverse Mobius transform of A114810. - Gary W. Adamson, Feb 28 2009

a(m) = m iff m is squarefree, a(A005117(n)) = A005117(n). - Reinhard Zumkeller, Mar 11 2012

Apostol & Tóth call this ϱ(n), i.e., varrho(n). - Charles R Greathouse IV, Apr 23 2013

References

J. Morgado, Inteiros regulares módulo n, Gazeta de Matematica (Lisboa), 33 (1972), no. 125-128, 1-5. [From Laszlo Toth, Sep 04 2008]

J. Morgado, A property of the Euler phi-function concerning the integers which are regular modulo n, Portugal. Math., 33 (1974), 185-191.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)

Osama Alkam and Emad Abu Osba, On the regular elements in Zn, Turk J Math, 32 (2008), 31-39.

B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.

Brăduţ Apostol and László Tóth, Some remarks on regular integers modulo n, arXiv:1304.2699 [math.NT], 2013.

Klaus Dohmen, On the Number of Regular Elements in Zn, arXiv:2304.02471 [math.CO], 2023.

S. R. Finch, Idempotents and nilpotents modulo n, arXiv:1304.2699 [math.NT], 2013.

V. S. Joshi, Order-free integers (mod m), Number Theory (Mysore, 1981), Lect. Notes in Math. 938, Springer-Verlag, 1982, pp. 93-100.

Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^2 / 12) for n = 1..100000

L. Tóth, Regular integers modulo n, arXiv:0710.1936 [math.NT], 2007-2008; Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275.

L. Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.

Formula

If n = product p_i^e_i, a(n) = product (1+p_i^e_i-p_i^(e_i-1)). - Vladeta Jovovic, Apr 19 2001

Dirichlet g.f.: zeta(s)*zeta(s-1)*product_{primes p} (1+p^(-2s)-p^(1-2s)-p^(-s)). - R. J. Mathar, Oct 24 2011

Dirichlet convolution square of A318661(n)/A318662(n). - Antti Karttunen, Sep 03 2018

Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.535896... - Vaclav Kotesovec, Dec 17 2019

Example

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.
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.

Maple

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:

Mathematica

a[n_] := Total[EulerPhi[Select[Divisors[n], GCD[#, n/#] == 1 &]]]; Array[a, 73] (* Jean-François Alcover, May 03 2011 *)
f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)

Prog

(Haskell)
a055653 = sum . map a000010 . a077610_row
-- Reinhard Zumkeller, Mar 11 2012
(PARI) a(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ Charles R Greathouse IV, Feb 19 2013, corrected by Antti Karttunen, Sep 03 2018
(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], f[i, 1]^f[i, 2]-f[i, 1]^(f[i, 2]-1)+1) \\ Charles R Greathouse IV, Feb 19 2013

Crossrefs

Cf. A000010, A053570, A053571, A000188, A006833, A055654, A157361, A114810, A000010, A077610, A318661, A318662.

Sequence in context: A353832 A085314 A085310 * A155918 A344705 A331170

Adjacent sequences: A055650 A055651 A055652 * A055654 A055655 A055656

Keyword

nonn,easy,nice,mult

Author

Labos Elemer, Jun 07 2000

Status

approved