A023896 Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention.

1, 1, 3, 4, 10, 6, 21, 16, 27, 20, 55, 24, 78, 42, 60, 64, 136, 54, 171, 80, 126, 110, 253, 96, 250, 156, 243, 168, 406, 120, 465, 256, 330, 272, 420, 216, 666, 342, 468, 320, 820, 252, 903, 440, 540, 506, 1081, 384, 1029, 500, 816, 624, 1378, 486, 1100, 672

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• easy
• nice

Comments

Sum of totatives of n, i.e., sum of integers up to n and coprime to n.

a(1) = 1, since 1 is coprime to any positive integer.

Row sums of A038566. - Wolfdieter Lang, May 03 2015

Islam & Manzoor prove that a(n) is an injection for n > 1, see links. In other words, if a(m) = a(n), and min(m, n) > 1, then m = n. - Muhammed Hedayet, May 19 2024

Examples

			G.f. = x + x^2 + 3*x^3 + 4*x^4 + 10*x^5 + 6*x^6 + 21*x^7 + 16*x^8 + 27*x^9 + ...
a(12) = 1 + 5 + 7 + 11 = 24.
n = 40: The smallest positive reduced residue system modulo 40 is {1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39}. The sum is a(40) = 320. Average is 20.
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).
• David M. Burton, Elementary Number Theory, p. 171.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2001, p. 163.
• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 111.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
• Geoffrey B. Campbell, Dirichlet summations and products over primes, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)
• Muhammed H. Islam and Shahriar Manzoor, φ1 and phitorial are injections, for any positive integer N, where N > 1.
• Constantin M. Petridi, The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots, arXiv:1612.07632 [math.NT], 2016.
• David Zmiaikou, Origamis and permutation groups, Thesis, 2011. See p. 65.

Crossrefs

Cf. A000010, A000203, A002180, A045545, A001783, A024816, A066760, A054521, A067392, A038566.

Row sums of A127368, A144734, A144824.

Cf. A000217, A002618, A008683, A009195, A023022, A065484, A076512, A175505, A175506.

Programs

Haskell

a023896 = sum . a038566_row  -- Reinhard Zumkeller, Mar 04 2012

Magma

[1] cat [n*EulerPhi(n)/2: n in [2..70]]; // Vincenzo Librandi, May 16 2015

Maple

A023896 := proc(n)
    if n = 1 then
        1;
    else
        n*numtheory[phi](n)/2 ;
    end if;
end proc: # R. J. Mathar, Sep 26 2013

Mathematica

a[ n_ ] = n/2*EulerPhi[ n ]; a[ 1 ] = 1; Table[a[n], {n, 56}]
a[ n_] := If[ n < 2, Boole[n == 1], Sum[ k Boole[1 == GCD[n, k]], { k, n}]]; (* Michael Somos, Jul 08 2014 *)

PARI

{a(n) = if(n<2, n>0, n*eulerphi(n)/2)};

PARI

A023896(n)=n*eulerphi(n)\/2 \\ about 10% faster. - M. F. Hasler, Feb 01 2021

Python

from sympy import totient
def A023896(n): return 1 if n == 1 else n*totient(n)//2 # Chai Wah Wu, Apr 08 2022

SageMath

def A023896(n): return 1 if n == 1 else n*euler_phi(n)//2
print([A023896(n) for n in range(1, 57)])  # Peter Luschny, Dec 03 2023

Formula

a(n) = n*A023022(n) for n > 2.

a(n) = phi(n^2)/2 = n*phi(n)/2 = A002618(n)/2 if n > 1, a(1)=1. See the Apostol reference for this exercise.

a(n) = Sum_{1 <= k < n, gcd(k, n) = 1} k.

If n = p is a prime, a(p) = T(p-1) where T(k) is the k-th triangular number (A000217). - Robert G. Wilson v, Jul 31 2004

Equals A054521 * [1,2,3,...]. - Gary W. Adamson, May 20 2007

a(n) = A053818(n) * A175506(n) / A175505(n). - Jaroslav Krizek, Aug 01 2010

If m,n > 1 and gcd(m,n) = 1 then a(m*n) = 2*a(m)*a(n). - Thomas Ordowski, Nov 09 2014

G.f.: Sum_{n>=1} mu(n)*n*x^n/(1-x^n)^3, where mu(n) = A008683(n). - Mamuka Jibladze, Apr 24 2015

G.f. A(x) satisfies A(x) = x/(1 - x)^3 - Sum_{k>=2} k * A(x^k). - Ilya Gutkovskiy, Sep 06 2019

For n > 1: a(n) = (n*A076512(n)/2)*A009195(n). - Jamie Morken, Dec 16 2019

Sum_{n>=1} 1/a(n) = 2 * A065484 - 1 = 3.407713... . - Amiram Eldar, Oct 09 2023

Extensions

Typos in programs corrected by Zak Seidov, Aug 03 2010

Name and example edited by Wolfdieter Lang, May 03 2015