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Search: id:A023896
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| A023896 |
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Sum of positive integers in reduced residue system modulo n. a(1) = 1 by convention. |
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+0
13
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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a(n) = Sum_{1<=k<=n, GCD(k,n)=1} k.
a(n) = n*A023022(n) for n>2.
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REFERENCES
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
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a(n)=n*phi(n)/2 if n>1, a(1)=0.
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 (qntmpkt(AT)yahoo.com), May 20 2007
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EXAMPLE
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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
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MATHEMATICA
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a[ n_ ]=n/2*EulerPhi[ n ]; a[ 1 ]=0.
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PROGRAM
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(PARI) a(n)=if(n<2, 0, n*eulerphi(n)/2)
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CROSSREFS
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Cf. A000010, A000203, A002180, A045545, A001783, A024816, A066760.
Cf. A054521.
Adjacent sequences: A023893 A023894 A023895 this_sequence A023897 A023898 A023899
Sequence in context: A073015 A063930 A014411 this_sequence A128488 A117781 A075100
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Olivier Gerard (ogerard(AT)ext.jussieu.fr)
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