A053191 -id:A053191 - cp's OEIS frontend

A239615 a(n) = n * A239612(n) / A053191(n).

1, 4, 5, 14, 11, 20, 13, 40, 21, 44, 21, 70, 27, 52, 55, 104, 35, 84, 37, 154, 65, 84, 45, 200, 85, 108, 81, 182, 59, 220, 61, 256, 105, 140, 143, 294, 75, 148, 135, 440, 83, 260, 85, 294, 231, 180, 93, 520, 133, 340, 175, 378, 107, 324, 231, 520, 185, 236

Offset: 1

José María Grau Ribas, Jun 25 2014

Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

Multiplicative because both A239612 and A053191 are. - Andrew Howroyd, Aug 07 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000
C. Calderón, J. M. Grau, A. Oller-Marcen, L. Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Cf. A239611, A239612, A239613, A239614, A079458, A053191, A227499.

a[n_] := Sum[Boole[GCD[x^2 + y^2 + z^2, n] == 1] GCD[x^2 + y^2 + z^2 - 1, n], {x, 1, n}, {y, 1, n}, {z, 1, n}]/(n EulerPhi[n]);
Array[a, 60] (* Jean-François Alcover, Nov 22 2018 *)

a(n)={my(p=lift(Mod(sum(i=0, n-1, x^(i^2%n)), x^n-1)^3)); sum(i=0, n-1, if(gcd(i,n)==1, polcoeff(p,i)*gcd((i-1)%n,n)))/(n * eulerphi(n))} \\ Andrew Howroyd, Aug 07 2018

A076669 Primes of the form n^2*totient(n)+1 (or A053191(n) + 1).

Original entry on oeis.org

2, 5, 19, 101, 73, 257, 487, 401, 577, 2029, 1801, 23549, 29401, 25601, 21169, 77659, 83233, 52489, 121001, 57601, 115321, 87121, 197137, 564899, 462401, 697049, 649801, 403369, 588061, 1020101, 887113, 893041, 922561, 696961, 1910413, 837937, 940801, 1411481

Offset: 1

Enoch Haga, Jan 16 2004

These primes are neither sorted nor unique. They are listed in the order they are found. - Michel Marcus, Jul 23 2015

Cf. A053191.

lista(nn) = for (n=1, nn, if (isprime(p=n^2*eulerphi(n)+1), print1(p, ", "))); \\ Michel Marcus, Jul 23 2015

Name corrected by Michel Marcus, Jul 23 2015

A000010 Euler totient function phi(n): count numbers <= n and prime to n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

			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)

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.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 409.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 154-156.
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.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B36.
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.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 2 vols., 1976, Vol. II, problem 71, p. 126.
Paulo Ribenboim, The New Book of Prime Number Records.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 28-33.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 162-167.

Daniel Forgues, Table of n, phi(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
Dario A. 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).
Alexander Bogomolny, Euler Function and Theorem.
Chris K. Caldwell, The Prime Glossary, Euler's phi function
Robert 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]
Kevin 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.
Derrick 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.
Keith 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 Erdős 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.
Wacław F. 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.
Ulrich Sondermann, Euler's Totient Function.
William 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.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
G. Villemin, Totient d'Euler.
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).
Gang Xiao, Numerical Calculator, To display phi(n) operate on "eulerphi(n)".
Index entries for "core" sequences
Index to divisibility sequences

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), A306633 (Dgf at s=3).

Axiom
```
[eulerPhi(n) for n in 1..100]
```

a n = length (filter (==1) (map (gcd n) [1..n])) -- Allan C. Wechsler, Dec 29 2014

# Computes the first N terms of the sequence.
function A000010List(N)
    phi = [i for i in 1:N + 1]
    for i in 2:N + 1
        if phi[i] == i
            for j in i:i:N + 1
                phi[j] -= div(phi[j], i)
    end end end
return phi end
println(A000010List(68))  # Peter Luschny, Sep 03 2023

[ EulerPhi(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

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
# Alternative without library function:
A000010List := proc(N) local i, j, phi;
    phi := Array([seq(i, i = 1 .. N+1)]);
    for i from 2 to N + 1 do
        if phi[i] = i then
            for j from i by i to N + 1 do
                phi[j] := phi[j] - iquo(phi[j], i) od
        fi od;
return phi end:
A000010List(68);  # Peter Luschny, Sep 03 2023

Mathematica
```
Array[EulerPhi, 70]
```

makelist(totient(n),n,0,1000); /* Emanuele Munarini, Mar 26 2011 */

{a(n) = if( n==0, 0, eulerphi(n))}; /* Michael Somos, Feb 05 2011 */

from sympy.ntheory import totient
print([totient(i) for i in range(1, 70)])  # Indranil Ghosh, Mar 17 2017

# Note also the implementation in A365339.

def A000010(n): return euler_phi(n) # Jaap Spies, Jan 07 2007

[euler_phi(n) for n in range(1, 70)]  # Zerinvary Lajos, Jun 06 2009

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)
a(n) = Sum_{d|n, e|n} gcd(d, e)*mobius(n/d)*mobius(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value p^e - p^(e-1) for n = p^e, a prime power). - Peter Bala, Jan 22 2024
Sum_{n >= 1} phi(n)*x^n/(1 + x^n) = x + 3*x^3 + 5*x^5 + 7*x^7 + ... = Sum_{n >= 1} phi(2*n-1)*x^(2*n-1)/(1 - x^(4*n-2)). For the first equality see Pólya and Szegő, problem 71, p. 126. - Peter Bala, Feb 29 2024
Conjecture: a(n) = lim_{k->oo} (n^(k + 1))/A000203(n^k). - Velin Yanev, Dec 04 2024 [A000010(p) = p-1, A000203(p^k) = (p^(k+1)-1)/(p-1), so the conjecture is true if n is prime. - Vaclav Kotesovec, Dec 19 2024]

A002618 a(n) = n*phi(n).

Original entry on oeis.org

1, 2, 6, 8, 20, 12, 42, 32, 54, 40, 110, 48, 156, 84, 120, 128, 272, 108, 342, 160, 252, 220, 506, 192, 500, 312, 486, 336, 812, 240, 930, 512, 660, 544, 840, 432, 1332, 684, 936, 640, 1640, 504, 1806, 880, 1080, 1012, 2162, 768, 2058, 1000

Offset: 1

N. J. A. Sloane

Also Euler phi function of n^2.
For n >= 3, a(n) is also the size of the automorphism group of the dihedral group of order 2n. This automorphism group is isomorphic to the group of transformations x -> ax + b, where a, b and x are integers modulo n and a is coprime to n. Its order is n*phi(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 18 2001
Order of metacyclic group of polynomial of degree n. - Artur Jasinski, Jan 22 2008
It appears that this sequence gives the number of permutations of 1, 2, 3, ..., n that are arithmetic progressions modulo n. - John W. Layman, Aug 27 2008
The conjecture by Layman is correct. Obviously any such permutation must have an increment that is prime to n, and almost as obvious that any such increment will work, with any starting value; hence phi(n) * n total. - Franklin T. Adams-Watters, Jun 09 2009
Consider the numbers from 1 to n^2 written line by line as an n X n square: a(n) = number of numbers that are coprime to all their horizontal and vertical immediate neighbors. - Reinhard Zumkeller, Apr 12 2011
n -> a(n) is injective: a(m) = a(n) implies m = n. - Franz Vrabec, Dec 12 2012 (See Mathematics Stack Exchange link for a proof.)
a(p) = p*(p-1) a pronic number, see A036689 and A002378. - Fred Daniel Kline, Mar 30 2015
Conjecture: All the rational numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
From Jianing Song, Aug 25 2023: (Start)
a(n) is the order of the holomorph (see the Wikipedia link) of the cyclic group of order n. Note that Hol(C_n) and Aut(D_{2n}) are isomorphic unless n = 2, where D_{2n} is the dihedral group of order 2*n. See the Wordpress link.
The odd-indexed terms form a subsequence of A341298: the holomorph of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link. (End)

			a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.

Peter Giblin, Primes and Programming: An Introduction to Number Theory with Computing. Cambridge: Cambridge University Press (1993) p. 116, Exercise 1.10.
J. L. Lagrange, Oeuvres, Vol. III Paris 1869.
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).

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Daniel Fischer, answer to Injectivity of the function n times the Euler Totient of n, Mathematics Stack Exchange, October 2013.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Dmitry Krachun and Zhi-Wei Sun, Each positive rational number has the form phi(m^2)/phi(n^2), arXiv:2001.03736 [math.HO], 2020. See also The American Mathematical Monthly (2020) Vol. 127, Issue 9, 847-849.
F. Luca and A. O. Munagi, The number of permutations which form arithmetic progressions modulo m, Annals of the Alexandru Ioan Cuza University, 2014, DOI: 10.2478/aicu-2014-0053. [Broken link]
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Indag. Math. Proc. Ser. A, 60. (1957) 608-619.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]
Wikipedia, Holomorph.
Wordpress, Automorphisms of dihedral groups.

First column of A047916.
Cf. A002619, A011755 (partial sums), A047918, A000010, A053650, A053191, A053192, A036689, A058161, A009262, A082473 (same terms, sorted into ascending order), A256545, A327172 (a left inverse), A327173, A065484.
Subsequence of A323333.

a002618 n = a000010 n * n  -- Reinhard Zumkeller, Dec 21 2012

[n*EulerPhi(n): n in [1..150]]; // Vincenzo Librandi, Apr 04 2011

with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007

Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)

numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

a(n)=n*eulerphi(n) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import totient as phi
def a(n): return n*phi(n)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 16 2022

[euler_phi(n^2) for n in range(1,51)] # Zerinvary Lajos, Jun 06 2009

Multiplicative with a(p^e) = (p-1)*p^(2e-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f.: zeta(s-2)/zeta(s-1). - R. J. Mathar, Feb 09 2011
a(n) = A173557(n) * A102631(n). - R. J. Mathar, Mar 30 2011
From Wolfdieter Lang, May 12 2011: (Start)
a(n)/2 = A023896(n), n >= 2.
a(n)/2 = (1/n) * Sum_{k=1..n-1, gcd(k,n)=1} k, n >= 2 (see A023896 and A076512/A109395). (End)
a(n) = lcm(phi(n^2),n). - Enrique Pérez Herrero, May 11 2012
a(n) = phi(n^2). - Wesley Ivan Hurt, Jun 16 2013
a(n) = A009195(n) * A009262(n). - Michel Marcus, Oct 24 2013
G.f.: -x + 2*Sum_{k>=1} mu(k)*k*x^k/(1 - x^k)^3. - Ilya Gutkovskiy, Jan 03 2017
a(n) = A082473(A327173(n)), A327172(a(n)) = n. -- Antti Karttunen, Sep 29 2019
Sum_{n>=1} 1/a(n) = 2.203856... (A065484). - Amiram Eldar, Sep 30 2019
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ c*sqrt(x) for c = Product_{p prime} (1 + 1/(p*(p - 1 + sqrt(p^2 - p)))) = 1.3651304521525857... - Charles R Greathouse IV, Mar 16 2022
a(n) = Sum_{d divides n} A001157(d)*A046692(n/d); that is, the Dirichlet convolution of sigma_2(n) and the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024

Better description from Labos Elemer, Feb 18 2000

A079458 Number of Gaussian integers in a reduced system modulo n.

Original entry on oeis.org

1, 2, 8, 8, 16, 16, 48, 32, 72, 32, 120, 64, 144, 96, 128, 128, 256, 144, 360, 128, 384, 240, 528, 256, 400, 288, 648, 384, 784, 256, 960, 512, 960, 512, 768, 576, 1296, 720, 1152, 512, 1600, 768, 1848, 960, 1152, 1056, 2208, 1024, 2352, 800, 2048, 1152, 2704

Offset: 1

Vladeta Jovovic, Jan 14 2003

Number of units in the ring consisting of the Gaussian integers modulo n. - Jason Kimberley, Dec 07 2015

			{1, i, 1+2i, 2+i, 3, 3i, 3+2i, 2+3i} is the set of eight units in the Gaussian integers modulo 4. - _Jason Kimberley_, Dec 07 2015

Jason Kimberley, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Wayback Machine link]
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Catalina Calderón, Jose Maria Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler's totient function, Publicationes Mathematicae-Debrecen, Vol. 87 (1-2) (2015), pp. 133-145; arXiv preprint, arXiv:1403.7878 [math.NT], 2014.

Equals four times A218147. - Jason Kimberley, Nov 14 2015
Sequences giving the number of solutions to the equation GCD(x_1^2+...+x_k^2, n) = 1 with 0 < x_i <= n: A000010 (k=1), A079458 (k=2), A053191 (k=3), A227499 (k=4), A238533 (k=5), A238534 (k=6), A239442 (k=7), A239441 (k=8), A239443 (k=9).
Cf. A003557, A007947, A204617, A062570.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), this sequence ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319445.

A079458 := func)>; // Jason Kimberley, Nov 14 2015

with(GaussInt): seq(GIphi(n), n=1..100);

phi[1]=1;phi[p_, s_] := Which[Mod[p, 4] == 3, p^(2 s - 2) (p^2 - 1), Mod[p, 4] == 1, p^(2 s - 2) ((p - 1))^2, True, 2^(2 s - 1)];phi[n_] := Product[phi[FactorInteger[n][[i, 1]], FactorInteger[n][[i, 2]]], {i, Length[FactorInteger[n]]}];Table[phi[n], {n, 1, 33}] (* José María Grau Ribas, Mar 16 2014 *)
f[p_, e_] := (p - 1)*p^(2*e - 1) * If[p == 2, 1, 1 - (-1)^((p-1)/2)/p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 13 2024 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, r*=2^(2*e-1));
        if(p%4==1, r*=(p-1)^2*p^(2*e-2));
        if(p%4==3, r*=(p^2-1)*p^(2*e-2));
    );
    return(r);
} \\ Jianing Song, Sep 16 2018

Multiplicative with a(2^e) = 2^(2*e-1), a(p^e) = (p^2-1)*p^(2*e-2) if p mod 4=3 and a(p^e) = (p-1)^2*p^(2*e-2) if p mod 4=1.
a(n) = A003557(n)^2 * a(A007947(n)), where a(2)=2, a(p)=(p-1)^2 for prime p=1(mod 4), a(p)=p^2-1 for prime p=3(mod 4), and a(n*m)=a(n)*a(m) for n coprime to m. - Jason Kimberley, Nov 16 2015
From Amiram Eldar, Feb 13 2024: (Start)
Dirichlet g.f.: zeta(s-2) * (1 - 1/2^(s-1)) * Product_{p prime > 2} (1 - 1/p^(s-1) - (-1)^((p-1)/2)*(p-1)/p^s).
Sum_{k=1..n} a(k) = c * n^3 / 3 + O(n^2 * log(n)), where c = (3/4) * Product_{p prime > 2} (1 - 1/p^2 - (-1)^((p-1)/2)*(p-1)/p^3) = (3/4) * A334427 * Product_{p prime == 1 (mod 4)} (1 - 2/p^2 + 1/p^3) = 0.6498027559... (Calderón et al., 2015). (End)
a(n) = A204617(n)*A062570(n). - Ridouane Oudra, Jun 05 2024

A238533 Number of solutions to gcd(x^2 + y^2 + z^2 + t^2 + h^2, n) = 1 with x,y,z,t,h in [0,n-1].

Original entry on oeis.org

1, 16, 162, 512, 2500, 2592, 14406, 16384, 39366, 40000, 146410, 82944, 342732, 230496, 405000, 524288, 1336336, 629856, 2345778, 1280000, 2333772, 2342560, 6156502, 2654208, 7812500, 5483712, 9565938, 7375872, 19803868, 6480000, 27705630, 16777216, 23718420

Offset: 1

José María Grau Ribas, Feb 28 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000
Param Parekh, Paavan Parekh, Sourav Deb, and Manish K. Gupta, On the Classification of Weierstrass Elliptic Curves over Z_n, arXiv:2310.11768 [cs.CR], 2023. See p. 9.

Cf. A079458, A227499, A238534.

Cf. n^k * a(n): A000010 (k=-4), A002618 (k=-3), A053191 (k=-2), A189393 (k=-1), A239442 (k=2), A239443 (k=4).

g[n_, 5] := g[n, 5] = Sum[If[GCD[x^2 + y^2 + z^2 + t^2 + h^2, n] == 1, 1, 0], {x, n}, {y, n}, {z, n}, {t, n}, {h, n}];Table[g[n,5] , {n, 1, 15}]
Table[n^4 * EulerPhi[n], {n, 1, 33}] (* Amiram Eldar, Dec 06 2020 *)

From Álvar Ibeas, Nov 24 2017: (Start)
a(n) = phi(n^5) = n^4 * phi(n), where phi=A000010.
Dirichlet g.f.: zeta(s - 5) / zeta(s - 4). The n-th term of the Dirichlet inverse is n^4 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221.
(End)
Sum_{k=1..n} a(k) ~ n^6 / Pi^2. - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^6 - p^5 - p + 1)) = 1.07162935672651489627... - Amiram Eldar, Dec 06 2020

A239442 a(n) = phi(n^7).

Original entry on oeis.org

1, 64, 1458, 8192, 62500, 93312, 705894, 1048576, 3188646, 4000000, 17715610, 11943936, 57921708, 45177216, 91125000, 134217728, 386201104, 204073344, 846825858, 512000000, 1029193452, 1133799040, 3256789558, 1528823808, 4882812500, 3706989312, 6973568802, 5782683648, 16655052988

Offset: 1

José María Grau Ribas, Mar 19 2014

Number of solutions of the equation gcd(x_1^2 + ... + x_7^2, n)=1 with 0 < x_i <= n.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Defining Phi_k(n):= number of solutions of the equation gcd(x_1^2 + ... + x_k^2, n) = 1 with 0 < x_i <= n.
Phi_1(n) = phi(n) = A000010.
Phi_2(n) = A079458.
Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191.
Phi_4(n) = A227499.
Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533.
Phi_6(n) = A238534.
Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442.
Phi_8(n) = A239441.
Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443.

with(numtheory); A239442:=n->phi(n^7); seq(A239442(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Mathematica
```
Table[EulerPhi[n^7], {n, 100}]
```

a(n) = n^6*eulerphi(n); \\ Michel Marcus, Mar 10 2018

a(n) = n^6*phi(n).
Dirichlet g.f.: zeta(s - 7) / zeta(s - 6). The n-th term of the Dirichlet inverse is n^6 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega=A001221. - Álvar Ibeas, Nov 24 2017
Sum_{k=1..n} a(k) ~ 3*n^8 / (4*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^8 - p^7 - p + 1)) = 1.01646280485545934937... - Amiram Eldar, Dec 06 2020

A239443 a(n) = phi(n^9), where phi = A000010.

Original entry on oeis.org

1, 256, 13122, 131072, 1562500, 3359232, 34588806, 67108864, 258280326, 400000000, 2143588810, 1719926784, 9788768652, 8854734336, 20503125000, 34359738368, 111612119056, 66119763456, 305704134738, 204800000000, 453874312332, 548758735360, 1722841676182, 880602513408, 3051757812500

Offset: 1

José María Grau Ribas, Mar 22 2014

Number of solutions of the equation GCD(x_1^2 + ... + x_9^2,n)=1 with 0 < x_i <= n.

In general, for m>0, Sum_{k=1..n} phi(k^m) ~ 6 * n^(m+1) / ((m+1)*Pi^2). - Vaclav Kotesovec, Feb 02 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000
C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Defining Phi_k(n):= number of solutions of the equation GCD(x_1^2 + ... + x_k^2,n)=1 with 0 < x_i <= n.
Phi_1(n) = phi(n) = A000010(n).
Phi_2(n) = A079458(n).
Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191(n).
Phi_4(n) = A227499(n).
Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533(n).
Phi_6(n) = A238534(n).
Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442(n).
Phi_8(n) = A239441(n).
Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443(n).

with(numtheory); A239443:=n->phi(n^9); seq(A239443(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Mathematica
```
Table[EulerPhi[n^9], {n, 100}]
```

a(n) = n^8*eulerphi(n); \\ Michel Marcus, Mar 10 2018

Dirichlet g.f.: zeta(s - 9) / zeta(s - 8). The n-th term of the Dirichlet inverse is n^8 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - Álvar Ibeas, Nov 24 2017
a(n) = n^8 * phi(n). - Altug Alkan, Mar 10 2018
Sum_{k=1..n} a(k) ~ 3*n^10 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^10 - p^9 - p + 1)) = 1.00399107654133714629... - Amiram Eldar, Dec 06 2020

A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 5, 23, 55, 155, 227, 521, 777, 1263, 1663, 2873, 3449, 5477, 6653, 8453, 10501, 15125, 17069, 23567, 26767, 32059, 36899, 48537, 53145, 65645, 73757, 86879, 96287, 119835, 127035, 155865, 172249, 194029, 212525, 241925, 257477, 306761, 332753, 369257

Offset: 1

Vaclav Kotesovec, Sep 10 2018

nonn

Comment from N. J. A. Sloane, Mar 22 2020: (Start)
Theorem: Sum_{ 1<=i<=n, 1<=j<=n, gcd(i,j)=1 } i*j = a(n).
Proof: From the Apostol reference we know that:
Sum_{ 1<=i<=n, gcd(i,n)=1 } i = n*phi(n)/2 (see A023896).
We use induction on n. The result is true for n=1.
Then a(n) - a(n-1) = 2*Sum_{ i=1..n-1, gcd(i,n)=1 } n*i = n^2*phi(n). QED (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A002088, A011755, A023896, A053191.

Accumulate[Table[k^2*EulerPhi[k], {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*eulerphi(k)); \\ Michel Marcus, Sep 12 2018

a(n) ~ 3*n^4 / (2*Pi^2).

A053192 a(n) is the cototient of n^3.

Original entry on oeis.org

0, 4, 9, 32, 25, 144, 49, 256, 243, 600, 121, 1152, 169, 1568, 1575, 2048, 289, 3888, 361, 4800, 3969, 5808, 529, 9216, 3125, 9464, 6561, 12544, 841, 19800, 961, 16384, 14157, 20808, 13475, 31104, 1369, 28880, 22815, 38400, 1681, 52920, 1849, 46464

Offset: 1

Labos Elemer, Mar 02 2000

For n^k, n^k - EulerPhi(n^k) = n^(k-1)*(n-EulerPhi(n)), or cototient(n^k) = n^(k-1)*cototient(n). A similar relation holds for Euler totient function.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A051953, A002618, A053650, A053191, A001248, A229099.

[n^3-EulerPhi(n^3): n in [1..44]]; // Vincenzo Librandi, Jul 28 2013

Table[(n^3 - EulerPhi[n^3]), {n, 1, 50}] (* Vincenzo Librandi, Jul 27 2013 *)

a(n) = n^3 - eulerphi(n^3) \\ Michel Marcus, Jul 26 2013

a(n) = n^2*Cototient(n) = A051953(n^3) = n^3 - EulerPhi(n^3) = Cototient(n^3).
a(prime(n)) = A051953(prime(n)^3) = A001248(n).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023

cp's OEIS Frontend

A239615 a(n) = n * A239612(n) / A053191(n).

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A076669 Primes of the form n^2*totient(n)+1 (or A053191(n) + 1).

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A000010 Euler totient function phi(n): count numbers <= n and prime to n.

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A002618 a(n) = n*phi(n).

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A079458 Number of Gaussian integers in a reduced system modulo n.

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A238533 Number of solutions to gcd(x^2 + y^2 + z^2 + t^2 + h^2, n) = 1 with x,y,z,t,h in [0,n-1].

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