|
| |
|
|
A023896
|
|
Sum of positive integers in reduced residue system modulo n. a(1) = 1 by convention.
|
|
41
| |
|
|
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; 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 (qntmpkt(AT)yahoo.com), Aug 27 2008]
Equals row sums of triangle A144734 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]
Equals row sums of triangle A144824 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 (jaroslav.krizek(AT)atlas.cz), 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 (qntmpkt(AT)yahoo.com), 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
|
|
|
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)
|
|
|
CROSSREFS
| Cf. A000010, A000203, A002180, A045545, A001783, A024816, A066760.
Cf. A054521, A067392.
Cf. A127368 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2008]
Cf. A144734 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]
Cf. A144824 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 21 2008]
Sequence in context: A143443 A139556 A191150 * A128488 A117781 A075100
Adjacent sequences: A023893 A023894 A023895 * A023897 A023898 A023899
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
|
|
|
EXTENSIONS
| Typos in programs corrected by Zak Seidov, Aug 03 2010
|
| |
|
|
