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A023896 Sum of positive integers in reduced residue system modulo n. a(1) = 1 by convention. 44
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

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

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.

Equals row sums of triangle A127368 [From Gary W. Adamson, Aug 27 2008]

Equals row sums of triangle A144734 [From Gary W. Adamson, Sep 20 2008]

Equals row sums of triangle A144824 [From Gary W. Adamson, Sep 21 2008]

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. [From Jaroslav Krizek, Aug 01 2010]

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.

Tattersall, J. "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, p. 163.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = n*phi(n)/2 = A002618(n)/2 if n>1, a(1)=1.

a(n) = Sum{1 <= k < n, k for GCD(k, n) =1}.

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

EXAMPLE

a(12) = 1 + 5 + 7 + 11 = 24.

Reduced residue system for 40 = {1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39}.  The sum is 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}]

PROG

(PARI) a(n)=if(n<2, 1, n*eulerphi(n)/2)

(Haskell)

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

CROSSREFS

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

Sequence in context: A143443 A139556 A191150 * A222136 A128488 A226303

Adjacent sequences:  A023893 A023894 A023895 * A023897 A023898 A023899

KEYWORD

nonn,easy,nice

AUTHOR

Olivier Gérard

EXTENSIONS

Typos in programs corrected by Zak Seidov, Aug 03 2010

STATUS

approved

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Last modified April 25 03:36 EDT 2014. Contains 240994 sequences.