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 A023896 Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention. 86
 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. a(n) = n*A023022(n) for n>2. a(n) = A053818(n) * A175506(n) / A175505(n). - Jaroslav Krizek, Aug 01 2010 Row sums of A038566. - Wolfdieter Lang, May 03 2015 REFERENCES T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n). D. M. Burton, Elementary Number Theory, p. 171. 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. 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. FORMULA 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 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 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 CROSSREFS Cf. A000010, A000203, A002180, A045545, A001783, A024816, A066760, A054521, A067392, A038566. Row sums of A127368, A144734, A144824. Cf. A023022. Sequence in context: A143443 A139556 A191150 * A328711 A343876 A356150 Adjacent sequences: A023893 A023894 A023895 * A023897 A023898 A023899 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 December 5 18:02 EST 2022. Contains 358588 sequences. (Running on oeis4.)