Search: a023022
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A023022
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Number of partitions of n into two relatively prime parts. After initial term, this is the "half-totient" function phi(n)/2 (A000010(n)/2).
(Formerly N0058)
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+40
77
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1, 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 4, 4, 8, 3, 9, 4, 6, 5, 11, 4, 10, 6, 9, 6, 14, 4, 15, 8, 10, 8, 12, 6, 18, 9, 12, 8, 20, 6, 21, 10, 12, 11, 23, 8, 21, 10, 16, 12, 26, 9, 20, 12, 18, 14, 29, 8, 30, 15, 18, 16, 24, 10, 33, 16, 22, 12, 35, 12, 36, 18, 20, 18, 30, 12, 39, 16, 27, 20, 41, 12
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OFFSET
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2,4
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COMMENTS
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The number of distinct linear fractional transformations of order n. Also the half-totient function can be used to construct a tree containing all the integers. On the zeroth rank we have just the integers 1 and 2: immediate "ancestors" of 1 and 2 are (1: 3,4,6 2: 5,8,10,12) etc. - Benoit Cloitre, Jun 03 2002
Moebius transform of floor(n/2). - Paul Barry, Mar 20 2005
Also number of different kinds of regular n-gons, one convex, the others self-intersecting. - Reinhard Zumkeller, Aug 20 2005
From Artur Jasinski, Oct 28 2008: (Start)
Degrees of minimal polynomials of cos(2*Pi/n). The first few are
1: x - 1
2: x + 1
3: 2*x + 1
4: x
5: 4*x^2 + 2*x - 1
6: 2*x - 1
7: 8*x^3 + 4*x^2 - 4*x - 1
8: 2*x^2 - 1
9: 8*x^3 - 6*x + 1
10: 4*x^2 - 2*x - 1
11: 32*x^5 + 16*x^4 - 32*x^3 - 12*x^2 + 6*x + 1
These polynomials have solvable Galois groups, so their roots can be expressed by radicals. (End)
a(n) is the number of rationals p/q in the interval [0,1] such that p + q = n. - Geoffrey Critzer, Oct 10 2011
It appears that, for n > 2, a(n) = A023896(n)/n. Also, it appears that a record occurs at n > 2 in this sequence if and only if n is a prime. For example, records occur at n=5, 7, 11, 13, 17, ..., all of which are prime. - John W. Layman, Mar 26 2012
From Wolfdieter Lang, Dec 19 2013: (Start)
a(n) is the degree of the algebraic number of s(n)^2 = (2*sin(Pi/n))^2, starting at a(1)=1. s(n) = 2*sin(Pi/n) is the length ratio side/R for a regular n-gon inscribed in a circle of radius R (in some length units). For the coefficient table of the minimal polynomials of s(n)^2 see A232633.
Because for even n, s(n)^2 lives in the algebraic number field Q(rho(n/2)), with rho(k) = 2*cos(Pi/k), the degree is a(2*l) = A055034(l). For odd n, s(n)^2 is an integer in Q(rho(n)), and the degree is a(2*l+1) = A055034(2*l+1) = phi(2*l+1)/2, l >= 1, with Euler's totient phi=A000010 and a(1)=1. See also A232631-A232633.
(End)
Also for n > 2: number of fractions A182972(k)/A182973(k) such that A182972(k) + A182973(k) = n, A182972(n) and A182973(n) provide an enumeration of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator. - Reinhard Zumkeller, Jul 30 2014
Number of distinct rectangles with relatively prime length and width such that L + W = n, W <= L. For a(17)=8; the rectangles are 1 X 16, 2 X 15, 3 X 14, 4 X 13, 5 X 12, 6 X 11, 7 X 10, 8 X 9. - Wesley Ivan Hurt, Nov 12 2017
After including a(1) = 1, the number of elements of any reduced residue system mod* n used by Brändli and Beyne is a(n). See the examples below. - Wolfdieter Lang, Apr 22 2020
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REFERENCES
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G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problems 60&61.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 2..10000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
K. S. Brown, The Half-Totient Tree
Tianxin Cai, Zhongyan Shen, and Mengjun Hu, On the Parity of the Generalized Euler Function, Advances in Mathematics (China), 2013, 42(4): 505-510.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Pinthira Tangsupphathawat, Takao Komatsu, Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
Eric Weisstein's World of Mathematics, Polygon Triangle Picking
Eric Weisstein's World of Mathematics, Trigonometry Angles
Canze Zhu and Qunying Liao, A recursion formula for the generalized Euler function phi_e(n), arXiv:2105.10870 [math.NT], 2021.
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FORMULA
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a(n) = phi(n)/2 for n >= 3.
a(n) = (1/n)*Sum_{k=1..n-1, gcd(n, k)=1} k = A023896(n)/n for n>2. - Reinhard Zumkeller, Aug 20 2005
G.f.: x*(x - 1)/2 + (1/2)*Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Apr 13 2017
a(n) = Sum_{d|n} moebius(n/d)*floor(d/2). - Michel Marcus, May 25 2021
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EXAMPLE
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a(15)=4 because there are 4 partitions of 15 into two parts that are relatively prime: 14 + 1, 13 + 2, 11 + 4, 8 + 7. - Geoffrey Critzer, Jan 25 2015
The smallest nonnegative reduced residue system mod*(n) for n = 1 is {0}, hence a(1) = 1; for n = 9 it is {1, 2, 4}, because 5 == 4 (mod* 9) since -5 == 4 (mod 9), 7 == 2 (mod* 9) and 8 == 1 (mod* 9). Hence a(9) = phi(9)/2 = 3. See the comment on Brändli and Beyne above. - Wolfdieter Lang, Apr 22 2020
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MAPLE
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A023022 := proc(n)
if n =2 then
1;
else
numtheory[phi](n)/2 ;
end if;
end proc:
seq(A023022(n), n=2..60) ; # R. J. Mathar, Sep 19 2017
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MATHEMATICA
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Join[{1}, Table[EulerPhi[n]/2, {n, 3, 100}]] (* adapted by Vincenzo Librandi, Aug 19 2018 *)
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PROG
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(PARI) a(n)=if(n<=2, 1, eulerphi(n)/2);
/* for printing minimal polynomials of cos(2*Pi/n) */
default(realprecision, 110);
for(n=1, 33, print(n, ": ", algdep(cos(2*Pi/n), a(n))));
(Haskell)
a023022 n = length [(u, v) | u <- [1 .. div n 2],
let v = n - u, gcd u v == 1]
-- Reinhard Zumkeller, Jul 30 2014
(Python)
from sympy.ntheory import totient
def a(n): return 1 if n<3 else totient(n)/2 # Indranil Ghosh, Mar 30 2017
(Magma) [1] cat [EulerPhi(n)/ 2: n in [3..100]]; // Vincenzo Librandi, Aug 19 2018
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CROSSREFS
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Cf. A000010, A055684, A046657, A049806, A049703, A062956.
Cf. A181875, A181876, A181877, A183918.
Cf. A023896.
Cf. A182972, A182973, A245497, A245718.
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane. This was in the 1973 "Handbook", but then was dropped from the database. Resubmitted by David W. Wilson.
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Jun 10 2012
Polynomials edited with the consent of Artur Jasinski by Wolfdieter Lang, Jan 08 2011
Name clarified by Geoffrey Critzer, Jan 25 2015
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STATUS
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approved
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A063534
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C(n) = H(n) + d(n), where C(n) is Chowla's function A048050, H(n) is the half-totient function A023022 and d(n) is the number of divisors function A000005.
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+20
2
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6, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753
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OFFSET
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1,1
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 1..1000
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FORMULA
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Conjecture: a(n) = A001748(n), n <> 2. - R. J. Mathar, Dec 15 2008
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PROG
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(PARI) C(n)=sigma(n)-n-1; H(n)=eulerphi(n)/2; j=[]; for(n=1, 1200, if(C(n)==H(n)+numdiv(n), j=concat(j, n))); j
(PARI) { n=0; for (m=1, 10^9, if (sigma(m) - m - 1 == eulerphi(m)/2 + numdiv(m), write("b063534.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009
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CROSSREFS
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Cf. A048050, A023022, A000005.
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KEYWORD
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easy,nonn
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AUTHOR
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Jason Earls, Aug 02 2001
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STATUS
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approved
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1, 1, 2, 3, 2, 2, 3, 4, 4, 3, 4, 6, 5, 4, 10, 6, 9, 6, 4, 8, 10, 8, 12, 6, 9, 12, 8, 6, 10, 12, 11, 8, 21, 10, 16, 12, 9, 20, 12, 18, 14, 8, 15, 18, 16, 24, 10, 16, 22, 12, 12, 18, 20, 18, 30, 12, 16, 27, 20, 12, 32, 21, 28, 20, 12, 36, 22, 30, 23, 36, 16, 21, 30, 20, 16, 24, 24, 26
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OFFSET
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1,3
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 1..1000
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PROG
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(PARI) H(n)=eulerphi(n)/2; j=[]; for(n=2, 200, if(isprime(n), n+1, j=concat(j, H(n) ))); j
(PARI) { n=0; for (m=2, 10^9, if (!isprime(m), write("b062948.txt", n++, " ", eulerphi(m)/2); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 14 2009
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CROSSREFS
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Cf. A023022, A002808.
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KEYWORD
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easy,nonn
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AUTHOR
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Jason Earls, Jul 21 2001
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EXTENSIONS
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Offset changed from 2 to 1 by Harry J. Smith, Aug 14 2009
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STATUS
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approved
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A062950
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C(H(n)), where C(n) is Chowla's function (A048050) and H(n) is the half-totient function (A023022).
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+20
1
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-1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 5, 0, 2, 2, 6, 0, 3, 2, 5, 0, 0, 2, 7, 5, 3, 5, 9, 2, 8, 6, 7, 6, 15, 5, 20, 3, 15, 6, 21, 5, 10, 7, 15, 0, 0, 6, 10, 7, 14, 15, 15, 3, 21, 15, 20, 9, 0, 6, 41, 8, 20, 14, 35, 7, 14, 14, 13, 15, 12, 15, 54, 20, 21, 20, 41, 15, 16, 14, 12, 21, 0, 15, 30, 10, 27, 21, 39, 15, 54, 13, 41, 0, 54, 14, 75, 10, 41, 21, 42, 14
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A062956
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a(n) = h(n^2) - h(n), where h(n) is the half-totient function (A023022).
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+20
1
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2, 3, 8, 5, 18, 14, 24, 18, 50, 22, 72, 39, 56, 60, 128, 51, 162, 76, 120, 105, 242, 92, 240, 150, 234, 162, 392, 116, 450, 248, 320, 264, 408, 210, 648, 333, 456, 312, 800, 246, 882, 430, 528, 495, 1058, 376, 1008, 490, 800, 612, 1352, 477, 1080, 660, 1008
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A063475
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Sum_{d | H(n)} d^2, where H(n) is the Half-Totient function (A023022).
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+20
1
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1, 1, 5, 1, 10, 5, 10, 5, 26, 5, 50, 10, 21, 21, 85, 10, 91, 21, 50, 26, 122, 21, 130, 50, 91, 50, 250, 21, 260, 85, 130, 85, 210, 50, 455, 91, 210, 85, 546, 50, 500, 130, 210, 122, 530, 85, 500, 130, 341, 210, 850, 91, 546, 210, 455, 250, 842, 85, 1300, 260, 455, 341
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OFFSET
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3,3
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 3..1000
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PROG
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(PARI) H(n)=eulerphi(n)/2; j=[]; for(n=3, 150, j=concat(j, sumdiv(H(n), d, d^2))); j
(PARI) { for (n=3, 1000, write("b063475.txt", n, " ", sumdiv(eulerphi(n)/2, d, d^2)) ) } \\ Harry J. Smith, Aug 22 2009
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CROSSREFS
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Cf. A023022.
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KEYWORD
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nonn
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AUTHOR
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Jason Earls, Jul 27 2001
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STATUS
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approved
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-1, 0, -2, 1, -3, 0, 0, 2, -5, 0, -6, 3, 4, 0, -8, 0, -9, 0, 6, 5, -11, 0, 0, 6, 0, 0, -14, -4, -15, 0, 10, 8, 12, 0, -18, 9, 12, 0, -20, -6, -21, 0, 0, 11, -23, 0, 0, 0, 16, 0, -26, 0, 20, 0, 18, 14, -29, 0, -30, 15, 0, 0, 24, -10, -33, 0, 22, -12, -35, 0, -36, 18, 0, 0, 30, -12, -39, 0, 0, 20, -41, 0, 32, 21, 28, 0, -44, 0, 36, 0, 30, 23, 36
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A000010
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Euler totient function phi(n): count numbers <= n and prime to n.
(Formerly M0299 N0111)
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+10
3757
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1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44
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OFFSET
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1,3
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COMMENTS
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Number of elements in a reduced residue system modulo n.
Degree of the n-th cyclotomic polynomial (cf. A013595). - Benoit Cloitre, Oct 12 2002
Number of distinct generators of a cyclic group of order n. Number of primitive n-th roots of unity. (A primitive n-th root x is such that x^k is not equal to 1 for k = 1, 2, ..., n - 1, but x^n = 1.) - Lekraj Beedassy, Mar 31 2005
Also number of complex Dirichlet characters modulo n; Sum_{k=1..n} a(k) is asymptotic to (3/Pi^2)*n^2. - Steven Finch, Feb 16 2006
a(n) is the highest degree of irreducible polynomial dividing 1 + x + x^2 + ... + x^(n-1) = (x^n - 1)/(x - 1). - Alexander Adamchuk, Sep 02 2006, corrected Sep 27 2006
a(p) = p - 1 for prime p. a(n) is even for n > 2. For n > 2, a(n)/2 = A023022(n) = number of partitions of n into 2 ordered relatively prime parts. - Alexander Adamchuk, Jan 25 2007
Number of automorphisms of the cyclic group of order n. - Benoit Jubin, Aug 09 2008
a(n+2) equals the number of palindromic Sturmian words of length n which are "bispecial", prefix or suffix of two Sturmian words of length n + 1. - Fred Lunnon, Sep 05 2010
Suppose that a and n are coprime positive integers, then by Euler's totient theorem, any factor of n divides a^phi(n) - 1. - Lei Zhou, Feb 28 2012
If m has k prime factors, (p_1, p_2, ..., p_k), then phi(m*n) = (Product_{i=1..k} phi (p_i*n))/phi(n)^(k-1). For example, phi(42*n) = phi(2*n)*phi(3*n)*phi(7*n)/phi(n)^2. - Gary Detlefs, Apr 21 2012
Sum_{n>=1} a(n)/n! = 1.954085357876006213144... This sum is referenced in Plouffe's inverter. - Alexander R. Povolotsky, Feb 02 2013 (see A336334. - Hugo Pfoertner, Jul 22 2020)
The order of the multiplicative group of units modulo n. - Michael Somos, Aug 27 2013
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 30 2016
From Eric Desbiaux, Jan 01 2017: (Start)
a(n) equals the Ramanujan sum c_n(n) (last term on n-th row of triangle A054533).
a(n) equals the Jordan function J_1(n) (cf. A007434, A059376, A059377, which are the Jordan functions J_2, J_3, J_4, respectively). (End)
For n > 1, a(n) appears to be equal to the number of semi-meander solutions for n with top arches containing exactly 2 mountain ranges and exactly 2 arches of length 1. - Roger Ford, Oct 11 2017
a(n) is the minimum dimension of a lattice able to generate, via cut-and-project, the quasilattice whose diffraction pattern features n-fold rotational symmetry. The case n=15 is the first n > 1 in which the following simpler definition fails: "a(n) is the minimum dimension of a lattice with n-fold rotational symmetry". - Felix Flicker, Nov 08 2017
Number of cyclic Latin squares of order n with the first row in ascending order. - Eduard I. Vatutin, Nov 01 2020
a(n) is the number of rational numbers p/q >= 0 (in lowest terms) such that p + q = n. - Rémy Sigrist, Jan 17 2021
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving Dirichlet convolution of a(n) and some sequence h(n) can be derived using the following (n >= 1):
Sum_{d|n} phi(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k)) [see P. H. van der Kamp link] = Sum_{d|n} h(d)*phi(n/d) = Sum_{k=1..n} h(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). Similarly,
Sum_{d|n} phi(d)*h(d) = Sum_{k=1..n} h(n/gcd(n,k)) = Sum_{k=1..n} h(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
More generally,
Sum_{d|n} h(d) = Sum_{k=1..n} h(gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))/phi(n/gcd(n,k)).
In particular, for sequences involving the Möbius transform:
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where mu = A008683.
Use of gcd(n,k)*lcm(n,k) = n*k and phi(gcd(n,k))*phi(lcm(n,k)) = phi(n)*phi(k) provide further variations. (End)
From Richard L. Ollerton, Nov 07 2021: (Start)
Formulas for products corresponding to the sums above may found using the substitution h(n) = log(f(n)) where f(n) > 0 (for example, cf. formulas for the sum A018804 and product A067911 of gcd(n,k)):
Product_{d|n} f(n/d)^phi(d) = Product_{k=1..n} f(gcd(n,k)) = Product_{d|n} f(d)^phi(n/d) = Product_{k=1..n} f(n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
Product_{d|n} f(d)^phi(d) = Product_{k=1..n} f(n/gcd(n,k)) = Product_{k=1..n} f(gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
Product_{d|n} f(d) = Product_{k=1..n} f(gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(1/phi(n/gcd(n,k))),
Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))), where mu = A008683. (End)
a(n+1) is the number of binary words with exactly n distinct subsequences (when n > 0). - Radoslaw Zak, Nov 29 2021
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
M. Baake and U. Grimm, Aperiodic Order Vol. 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications 149, Cambridge University Press, 2013: see Tables 3.1 and 3.2.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
C. W. Curtis, Pioneers of Representation Theory ..., Amer. Math. Soc., 1999; see p. 3.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Paris, 2004, Problème 529, pp. 71-257.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, Chapter V.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 21.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, p. 137.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 60, 62, 63, 288, 323, 328, 330.
Peter Hilton and Jean Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, pages 261-264, the Coach theorem.
Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21 pp. 281-294.
P. Ribenboim, The New Book of Prime Number Records.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Daniel Forgues, Table of n, phi(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
D. Alpern, Factorization using the Elliptic Curve Method(along with sigma_0, sigma_1 and phi functions)
Joerg Arndt, Matters Computational (The Fxtbook), section 39.7, pp. 776-778.
F. Bayart, Indicateur d'Euler (in French).
A. Bogomolny, Euler Function and Theorem.
C. K. Caldwell, The Prime Glossary, Euler's phi function
R. D. Carmichael, A table of the values of m corresponding to given values of phi(m), Amer. J. Math., 30 (1908), 394-400. [Annotated scanned copy]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
K. Ford, The number of solutions of phi(x)=m, arXiv:math/9907204 [math.NT], 1999.
Kevin Ford, Florian Luca and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805 [math.NT], 2011.
H. Fripertinger, The Euler phi function.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
E. Pérez Herrero, Totient Carnival partitions, Psychedelic Geometry Blogspot.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, arXiv:1201.3139 [math.NT]
M. Lal and P. Gillard, Table of Euler's phi function, n < 10^5, Math. Comp., 23 (1969), 682-683.
D. N. Lehmer, Review of Dickson's History of the Theory of Numbers, Bull. Amer. Math. Soc., 26 (1919), 125-132.
Peter Luschny, Sequences related to Euler's totient function.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
Mathematics Stack Exchange, Is the Euler phi function bounded below? (2013).
Mathforum, Proving phi(m) Is Even.
K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n).
Graeme McRae, Euler's Totient Function.
François Nicolas, A simple, polynomial-time algorithm for the matrix torsion problem, arXiv:0806.2068 [cs.DM], 2009.
Matthew Parker, The first 5 million terms (7-Zip compressed file).
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Primefan, Euler's Totient Function Values For n=1 to 500, with Divisor Lists.
Marko Riedel, Combinatorics and number theory page.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.
K. Schneider, Euler phi-function, PlanetMath.org.
W. Sierpiński, Euler's Totient Function And The Theorem Of Euler.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 14.
U. Sondermann, Euler's Totient Function.
W. A. Stein, Phi is a Multiplicative Function
Pinthira Tangsupphathawat, Takao Komatsu and Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
G. Villemin, Totient d'Euler.
A. de Vries, The prime factors of an integer (along with Euler's phi and Carmichael's lambda functions)
K. W. Wegner, Values of phi(x) = n for n from 2 through 1978, mimeographed manuscript, no date. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.
Wikipedia, Multiplicative group of integers modulo n.
Wikipedia, Ramanujan's sum
Wolfram Research, First 50 values of phi(n).
G. Xiao, Numerical Calculator, To display phi(n) operate on "eulerphi(n)".
Index entries for "core" sequences
Index to divisibility sequences
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FORMULA
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phi(n) = n*Product_{distinct primes p dividing n} (1 - 1/p).
Sum_{d divides n} phi(d) = n.
phi(n) = Sum_{d divides n} mu(d)*n/d, i.e., the Moebius transform of the natural numbers; mu() = Moebius function A008683().
Dirichlet generating function Sum_{n>=1} phi(n)/n^s = zeta(s-1)/zeta(s). Also Sum_{n >= 1} phi(n)*x^n/(1 - x^n) = x/(1 - x)^2.
Multiplicative with a(p^e) = (p - 1)*p^(e-1). - David W. Wilson, Aug 01 2001
Sum_{n>=1} (phi(n)*log(1 - x^n)/n) = -x/(1 - x) for -1 < x < 1 (cf. A002088) - Henry Bottomley, Nov 16 2001
a(n) = binomial(n+1, 2) - Sum_{i=1..n-1} a(i)*floor(n/i) (see A000217 for inverse). - Jon Perry, Mar 02 2004
It is a classical result (certainly known to Landau, 1909) that lim inf n/phi(n) = 1 (taking n to be primes), lim sup n/(phi(n)*log(log(n))) = e^gamma, with gamma = Euler's constant (taking n to be products of consecutive primes starting from 2 and applying Mertens' theorem). See e.g. Ribenboim, pp. 319-320. - Pieter Moree, Sep 10 2004
a(n) = Sum_{i=1..n} |k(n, i)| where k(n, i) is the Kronecker symbol. Also a(n) = n - #{1 <= i <= n : k(n, i) = 0}. - Benoit Cloitre, Aug 06 2004 [Corrected by Jianing Song, Sep 25 2018]
Conjecture: Sum_{i>=2} (-1)^i/(i*phi(i)) exists and is approximately 0.558 (A335319). - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004
From Enrique Pérez Herrero, Sep 07 2010: (Start)
a(n) = Sum_{i=1..n} floor(sigma_k(i*n)/sigma_k(i)*sigma_k(n)), where sigma_2 is A001157.
a(n) = Sum_{i=1..n} floor(tau_k(i*n)/tau_k(i)*tau_k(n)), where tau_3 is A007425.
a(n) = Sum_{i=1..n} floor(rad(i*n)/rad(i)*rad(n)), where rad is A007947. (End)
a(n) = A173557(n)*A003557(n). - R. J. Mathar, Mar 30 2011
a(n) = A096396(n) + A096397(n). - Reinhard Zumkeller, Mar 24 2012
phi(p*n) = phi(n)*(floor(((n + p - 1) mod p)/(p - 1)) + p - 1), for primes p. - Gary Detlefs, Apr 21 2012
For odd n, a(n) = 2*A135303((n-1)/2)*A003558((n-1)/2) or phi(n) = 2*c*k; the Coach theorem of Pedersen et al. Cf. A135303. - Gary W. Adamson, Aug 15 2012
G.f.: Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2, where mu(n) = A008683(n). - Mamuka Jibladze, Apr 05 2015
a(n) = n - cototient(n) = n - A051953(n). - Omar E. Pol, May 14 2016
a(n) = lim_{s->1} n*zeta(s)*(Sum_{d divides n} A008683(d)/(e^(1/d))^(s-1)), for n > 1. - Mats Granvik, Jan 26 2017
Conjecture: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 for n > 1. The sum is over a,b,c such that n*c - a*b = 1. - Benedict W. J. Irwin, Apr 03 2017
a(n) = Sum_{j=1..n} gcd(j, n) cos(2*Pi*j/n) = Sum_{j=1..n} gcd(j, n) exp(2*Pi*i*j/n) where i is the imaginary unit. Notice that the Ramanujan's sum c_n(k) := Sum_{j=1..n, gcd(j, n) = 1} exp(2*Pi*i*j*k/n) gives a(n) = Sum_{k|n} k*c_(n/k)(1) = Sum_{k|n} k*mu(n/k). - Michael Somos, May 13 2018
G.f.: x*d/dx(x*d/dx(log(Product_{k>=1} (1 - x^k)^(-mu(k)/k^2)))), where mu(n) = A008683(n). - Mamuka Jibladze, Sep 20 2018
a(n) = Sum_{d|n} A007431(d). - Steven Foster Clark, May 29 2019
G.f. A(x) satisfies: A(x) = x/(1 - x)^2 - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019
a(n) >= sqrt(n/2) (Nicolas). - Hugo Pfoertner, Jun 01 2020
a(n) > n/(exp(gamma)*log(log(n)) + 5/(2*log(log(n)))), except for n=223092870 (Rosser, Schoenfeld). - Hugo Pfoertner, Jun 02 2020
From Bernard Schott, Nov 28 2020: (Start)
Sum_{m=1..n} 1/a(m) = A028415(n)/A048049(n) -> oo when n->oo.
Sum_{n >= 1} 1/a(n)^2 = A109695.
Sum_{n >= 1} 1/a(n)^3 = A335818.
Sum_{n >= 1} 1/a(n)^k is convergent iff k > 1.
a(2n) = a(n) iff n is odd, and, a(2n) > a(n) iff n is even. (End) [Actually, a(2n) = 2*a(n) for even n. - Jianing Song, Sep 18 2022]
a(n) = 2*A023896(n)/n, n > 1. - Richard R. Forberg, Feb 03 2021
From Richard L. Ollerton, May 09 2021: (Start)
For n > 1, Sum_{k=1..n} phi^{(-1)}(n/gcd(n,k))*a(gcd(n,k))/a(n/gcd(n,k)) = 0, where phi^{(-1)} = A023900.
For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(gcd(n,k)))*rad(gcd(n,k))/gcd(n,k) = 0.
For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(n/gcd(n,k)))*rad(n/gcd(n,k))*gcd(n,k) = 0.
Sum_{k=1..n} a(gcd(n,k))/a(n/gcd(n,k)) = n. (End)
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EXAMPLE
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G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 4*x^8 + 6*x^9 + 4*x^10 + ...
a(8) = 4 with {1, 3, 5, 7} units modulo 8. a(10) = 4 with {1, 3, 7, 9} units modulo 10. - Michael Somos, Aug 27 2013
From Eduard I. Vatutin, Nov 01 2020: (Start)
The a(5)=4 cyclic Latin squares with the first row in ascending order are:
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 4 0 1 2 3
2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2
3 4 0 1 2 1 2 3 4 0 4 0 1 2 3 2 3 4 0 1
4 0 1 2 3 3 4 0 1 2 2 3 4 0 1 1 2 3 4 0
(End)
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MAPLE
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with(numtheory): A000010 := phi; [ seq(phi(n), n=1..100) ]; # version 1
with(numtheory): phi := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := n*mul((1-1/t1[i][1]), i=1..nops(t1)); end; # version 2
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MATHEMATICA
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Array[EulerPhi, 70]
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PROG
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(Axiom) [eulerPhi(n) for n in 1..100]
(Magma) [ EulerPhi(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(PARI) {a(n) = if( n==0, 0, eulerphi(n))}; /* Michael Somos, Feb 05 2011 */
(Sage)
# euler_phi is a standard function in Sage.
def A000010(n): return euler_phi(n)
def A000010_list(n): return [ euler_phi(i) for i in range(1, n+1)]
# Jaap Spies, Jan 07 2007
(PARI) { for (n=1, 100000, write("b000010.txt", n, " ", eulerphi(n))); } \\ Harry J. Smith, Apr 26 2009
(Sage) [euler_phi(n) for n in range(1, 70)] # Zerinvary Lajos, Jun 06 2009
(Maxima) makelist(totient(n), n, 0, 1000); /* Emanuele Munarini, Mar 26 2011 */
(Haskell) a n = length (filter (==1) (map (gcd n) [1..n])) -- Allan C. Wechsler, Dec 29 2014
(Python)
from sympy.ntheory import totient
print([totient(i) for i in range(1, 70)]) # Indranil Ghosh, Mar 17 2017
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CROSSREFS
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Cf. A002088 (partial sums), A008683, A003434 (steps to reach 1), A007755, A049108, A002202 (values), A011755 (Sum k*phi(k)).
Cf. also A005277 (nontotient numbers). For inverse see A002181, A006511, A058277.
Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), this sequence (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A054521, A023022, A054525.
Row sums of triangles A134540, A127448, A143239, A143353 and A143276.
Equals right and left borders of triangle A159937. - Gary W. Adamson, Apr 26 2009
Values for prime powers p^e: A006093 (e=1), A036689 (e=2), A135177 (e=3), A138403 (e=4), A138407 (e=5), A138412 (e=6).
Values for perfect powers n^e: A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), A306411 (e=6), A239442 (e=7), A306412 (e=8), A239443 (e=9).
Cf. A003558, A135303.
Cf. A152455, A080737.
Cf. A076479.
Cf. A023900 (Dirichlet inverse of phi).
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KEYWORD
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easy,core,nonn,mult,nice,hear
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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A023896
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Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention.
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+10
86
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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;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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
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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.
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.
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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A023896 := proc(n)
if n = 1 then
1;
else
n*numtheory[phi](n)/2 ;
end if;
end proc: # R. J. Mathar, Sep 26 2013
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000010, A000203, A002180, A045545, A001783, A024816, A066760, A054521, A067392, A038566.
Row sums of A127368, A144734, A144824.
Cf. A023022.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Olivier Gérard
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EXTENSIONS
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Typos in programs corrected by Zak Seidov, Aug 03 2010
Name and example edited by Wolfdieter Lang, May 03 2015
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STATUS
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approved
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A220377
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Number of partitions of n into three distinct and mutually relatively prime parts.
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+10
38
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1, 0, 2, 1, 3, 1, 6, 1, 7, 3, 7, 3, 14, 3, 15, 6, 14, 6, 25, 6, 22, 10, 25, 9, 42, 8, 34, 15, 37, 15, 53, 13, 48, 22, 53, 17, 78, 17, 65, 30, 63, 24, 99, 24, 88, 35, 84, 30, 126, 34, 103, 45, 103, 38, 166, 35, 124, 57, 128, 51, 184, 44, 150, 67, 172, 52, 218
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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6,3
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COMMENTS
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The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020
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LINKS
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Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000 (terms 6..1000 from Seiichi Manyama)
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FORMULA
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a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020
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EXAMPLE
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For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
From Gus Wiseman, Oct 14 2020: (Start)
The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
321 . 431 531 532 731 543 751 743 753 754 971 765 B53 875
521 541 651 752 951 853 B51 873 B71 974
721 732 761 B31 871 D31 954 D51 A73
741 851 952 972 A91
831 941 B32 981 B54
921 A31 B41 A71 B72
B21 D21 B43 B81
B52 C71
B61 D43
C51 D52
D32 D61
D41 E51
E31 F41
F21 G31
H21
(End)
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MATHEMATICA
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Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
Table[Count[IntegerPartitions[n, {3}], _?(CoprimeQ@@#&&Length[ Union[#]] == 3&)], {n, 6, 100}] (* Harvey P. Dale, May 22 2020 *)
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PROG
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(PARI) a(n)=my(P=partitions(n)); sum(i=1, #P, #P[i]==3&&P[i][1]<P[i][2]&&P[i][2]<P[i][3]&&gcd(P[i][1], P[i][2])==1&&gcd(P[i][1], P[i][3])==1&&gcd(P[i][2], P[i][3])==1) \\ Charles R Greathouse IV, Dec 14 2012
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CROSSREFS
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Cf. A015617, A300815.
A023022 is the 2-part version.
A101271 is the relative prime instead of pairwise coprime version.
A220377*6 is the ordered version.
A305713 counts these partitions of any length, with Heinz numbers A302797.
A307719 is the non-strict version.
A337461 is the non-strict ordered version.
A337563 is the case with no 1's.
A337605 is the pairwise non-coprime instead of pairwise coprime version.
A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.
A008284 counts partitions by sum and length, with strict case A008289.
A318717 counts pairwise non-coprime strict partitions.
A326675 ranks pairwise coprime sets.
A327516 counts pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.
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KEYWORD
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nonn
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AUTHOR
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Carl Najafi, Dec 13 2012
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STATUS
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approved
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