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A023896 Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention. 90
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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
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.
David Zmiaikou, Origamis and permutation groups, Thesis, 2011. See p. 65.
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
EXAMPLE
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.
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 *)
PROG
(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
(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
(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
CROSSREFS
Row sums of A127368, A144734, A144824.
Sequence in context: A143443 A139556 A191150 * A328711 A361655 A343876
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Typos in programs corrected by Zak Seidov, Aug 03 2010
Name and example edited by Wolfdieter Lang, May 03 2015
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)