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 A023896 Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention. 65
 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). For number n >= 1 holds B(n) = a(n) / A023896(n) = A175505(n) / A175506(n), where B(n) = antiharmonic mean of numbers k such that GCD(k, n) = 1 for k < n. Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A007645, A003627, A034934. - 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. LINKS Michael De Vlieger (First 1000 terms from T. D. Noe), Table of n, a(n) for n = 1..10000 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 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)}; (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 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 * A222136 A279787 A128488 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 October 16 17:49 EDT 2019. Contains 328102 sequences. (Running on oeis4.)