A330523 -id:A330523 - cp's OEIS frontend

A062354 a(n) = sigma(n)*phi(n).

1, 3, 8, 14, 24, 24, 48, 60, 78, 72, 120, 112, 168, 144, 192, 248, 288, 234, 360, 336, 384, 360, 528, 480, 620, 504, 720, 672, 840, 576, 960, 1008, 960, 864, 1152, 1092, 1368, 1080, 1344, 1440, 1680, 1152, 1848, 1680, 1872, 1584, 2208, 1984, 2394, 1860

Offset: 1

Jason Earls, Jul 06 2001

Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the number of conjugacy classes in G_n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 13 2001
a(n) = Sum_{d|n} phi(n*d). - Vladeta Jovovic, Apr 17 2002
Apparently the Mobius transform of A062952. - R. J. Mathar, Oct 01 2011

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.

T. D. Noe, Table of n, a(n) for n=1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000
J.-L. Nicolas and Jonathan Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

Cf. A000010, A000203, A000252, A062355, A064840, A093827.

Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))

Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - Vladeta Jovovic, Apr 17 2002
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*product_{primes p} (1-p^(1-s)-p^(-s)+p^(2-2s)). - R. J. Mathar, Oct 01 2011, corrected by Vaclav Kotesovec, Dec 17 2019
6/Pi^2 < a(n)/n^2 < 1 for n > 1. - Jonathan Sondow, Mar 06 2014
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.535896... - Vaclav Kotesovec, Dec 17 2019
Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - Amiram Eldar, Aug 20 2020
a(n)/n^2 > 8/Pi^2 for odd n. - M. F. Hasler, Jul 08 2025

A000252 Number of invertible 2 X 2 matrices mod n.

Original entry on oeis.org

1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248, 1822176, 738720

Offset: 1

N. J. A. Sloane

For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the Product_{i=1..k} (p_i)^(e_i), then a(n) = Product_{i=1..k} (((p_i)^(2*e_i) - (p_i)^(2*e_i - 2)) * ((p_i)^(2*e_i) - (p_i)^(2*e_i - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
a(n) is the order of the automorphism group of the group C_n X C_n, where C_n is the cyclic group of order n. - Laszlo Toth, Dec 06 2011
Order of the group GL(2,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Jul 08 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.

The order of GL_2(K) for a finite field K is in sequence A059238.
Row n=2 of A316622.
Row sums of A316566.
Cf. A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A227499.

Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011, after Vladeta Jovovic *)

a(n)=my(f=factor(n)[,1]); n^4*prod(i=1,#f, (1-1/f[i]^2)*(1-1/f[i])) \\ Charles R Greathouse IV, Feb 06 2017

from math import prod
from sympy import factorint
def A000252(n): return prod(p**((e<<2)-3)*(p*(p*(p-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = n^4*Product_{primes p dividing n} (1 - 1/p^2)*(1 - 1/p) = n^4*Product_{primes p dividing n} p^(-3)*(p^2 - 1)*(p - 1). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
Multiplicative with a(p^e) = (p - 1)^2*(p + 1)*p^(4e-3). - David W. Wilson, Aug 01 2001
a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
Dirichlet g.f.: zeta(s - 4)*Product_{p prime} (1 - p^(1 - s)*(p^2 + p - 1)). - Álvar Ibeas, Nov 28 2017
a(n) = A227499(n) for odd n; (3/4)*A227499(n) for even n. - Jianing Song, Jul 08 2018
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Aug 20 2021
Sum_{n>=1} 1/a(n) = (Pi^8/3240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.2059016071... . - Amiram Eldar, Dec 03 2022

More terms from David W. Wilson, Jul 21 2001

A055653 Sum of phi(d) [A000010] over all unitary divisors d of n (that is, gcd(d,n/d) = 1).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 9, 17, 14, 19, 15, 21, 22, 23, 15, 21, 26, 19, 21, 29, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 46, 47, 27, 43, 42, 51, 39, 53, 38, 55, 35, 57, 58, 59, 45, 61, 62, 49, 33, 65, 66, 67, 51, 69, 70, 71, 35, 73

Offset: 1

Labos Elemer, Jun 07 2000

Phi-summation over d-s if runs over all divisors is n, so these values do not exceed n. Compare also other "Phi-summations" like A053570, A053571, or distinct primes dividing n, etc.
a(n) is also the number of solutions of x^(k+1)=x mod n for some k>=1. - Steven Finch, Apr 11 2006
An integer a is called regular (mod n) if there is an integer x such that a^2 x == a (mod n). Then a(n) is also the number of regular integers a (mod n) such that 1 <= a <= n. - Laszlo Toth, Sep 04 2008
Equals row sums of triangle A157361 and inverse Mobius transform of A114810. - Gary W. Adamson, Feb 28 2009
a(m) = m iff m is squarefree, a(A005117(n)) = A005117(n). - Reinhard Zumkeller, Mar 11 2012
Apostol & Tóth call this ϱ(n), i.e., varrho(n). - Charles R Greathouse IV, Apr 23 2013

			n=1260 has 36 divisors of which 16 are unitary ones: {1,4,5,7,9,20,28,35,36,45,63,140,180,252,315,1260}.
EulerPhi values of these divisors are: {1,2,4,6,6,8,12,24,12,24,36,48,48,72,144,288}.
The sum is 735, thus a(1260)=735.
Or, 1260=2^2*3^2*5*7, thus a(1260) = (1 + 2^2 - 2)*(1 + 3^2 - 3)*(1 + 5 - 5^0)*(1 + 7 - 7^0) = 735.

J. Morgado, Inteiros regulares módulo n, Gazeta de Matematica (Lisboa), 33 (1972), no. 125-128, 1-5. [From Laszlo Toth, Sep 04 2008]
J. Morgado, A property of the Euler phi-function concerning the integers which are regular modulo n, Portugal. Math., 33 (1974), 185-191.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Osama Alkam and Emad Abu Osba, On the regular elements in Zn, Turk J Math, 32 (2008), 31-39.
B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
Brăduţ Apostol and László Tóth, Some remarks on regular integers modulo n, arXiv:1304.2699 [math.NT], 2013.
Klaus Dohmen, On the Number of Regular Elements in Zn, arXiv:2304.02471 [math.CO], 2023.
S. R. Finch, Idempotents and nilpotents modulo n, arXiv:1304.2699 [math.NT], 2013.
V. S. Joshi, Order-free integers (mod m), Number Theory (Mysore, 1981), Lect. Notes in Math. 938, Springer-Verlag, 1982, pp. 93-100.
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^2 / 12) for n = 1..100000
Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025.
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See phi+ function.
L. Tóth, Regular integers modulo n, arXiv:0710.1936 [math.NT], 2007-2008; Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275.
L. Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.

Cf. A000010, A053570, A053571, A000188, A008833, A055654, A157361, A114810, A000010, A077610, A318661, A318662.

a055653 = sum . map a000010 . a077610_row
-- Reinhard Zumkeller, Mar 11 2012

A055653 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]-ifactors(n)[ 2 ][ i ] [ 1 ]^(ifactors(n)[ 2 ] [ i ] [ 2 ]-1)): od: RETURN(ans) end:

a[n_] := Total[EulerPhi[Select[Divisors[n], GCD[#, n/#] == 1 &]]]; Array[a, 73] (* Jean-François Alcover, May 03 2011 *)
f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)

a(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ Charles R Greathouse IV, Feb 19 2013, corrected by Antti Karttunen, Sep 03 2018

a(n)=my(f=factor(n));prod(i=1,#f[,1],f[i,1]^f[i,2]-f[i,1]^(f[i,2]-1)+1) \\ Charles R Greathouse IV, Feb 19 2013

If n = product p_i^e_i, a(n) = product (1+p_i^e_i-p_i^(e_i-1)). - Vladeta Jovovic, Apr 19 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*product_{primes p} (1+p^(-2s)-p^(1-2s)-p^(-s)). - R. J. Mathar, Oct 24 2011
Dirichlet convolution square of A318661(n)/A318662(n). - Antti Karttunen, Sep 03 2018
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.535896... - Vaclav Kotesovec, Dec 17 2019

A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

Original entry on oeis.org

1, 4, 9, 18, 25, 36, 49, 78, 87, 100, 121, 162, 169, 196, 225, 326, 289, 348, 361, 450, 441, 484, 529, 702, 645, 676, 807, 882, 841, 900, 961, 1334, 1089, 1156, 1225, 1566, 1369, 1444, 1521, 1950, 1681, 1764, 1849, 2178, 2175, 2116, 2209, 2934, 2443, 2580

Offset: 1

Vladeta Jovovic, Jul 21 2001

If k is squarefree (cf. A005117) then A062952(k) = k^2. - Benoit Cloitre, Apr 16 2002

Inverse Möbius transform of A062354(n). - Wesley Ivan Hurt, Jul 26 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A029939, A057660, A062952.
Cf. A002117, A013661, A183699, A330523.
Cf. A062354.

f[p_, e_] := (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{d|n} phi(d)*sigma(d).
a(n) = Sum_{k=1..n} sigma(n/gcd(n, k)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A183699 * A330523 / 3. - Amiram Eldar, Oct 30 2022

A256392 Decimal expansion of Product_{p prime} (1-4/p^2+4/p^3-1/p^4).

Original entry on oeis.org

2, 1, 7, 7, 7, 8, 7, 1, 6, 6, 1, 9, 5, 3, 6, 3, 7, 8, 3, 2, 3, 0, 0, 7, 5, 1, 4, 1, 1, 9, 4, 4, 6, 8, 1, 3, 1, 3, 0, 7, 9, 7, 7, 5, 5, 0, 0, 1, 3, 5, 5, 9, 3, 7, 6, 4, 8, 2, 7, 6, 4, 0, 3, 5, 2, 3, 6, 2, 6, 4, 9, 1, 1, 1, 2, 2, 5, 2, 6, 2, 0, 5, 5, 7, 9, 2, 5, 4, 4, 3, 8, 2, 3, 5, 6, 3, 7, 6, 5, 6, 9, 1, 8, 3, 3, 9

Offset: 0

Juan Arias-de-Reyna, Mar 28 2015

Also decimal expansion of the probability that an integer tuple (x,y,z,w) satisfies gcd(x,y) = gcd(y,z) = gcd(z,w) = gcd(w,x) = 1.

			0.2177787166195363783230075141...

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1000
Juan Arias de Reyna and Randell Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq., Vol. 18 (2015), Article 15.10.4; arXiv preprint, arXiv:1403.2769 [math.NT], 2014.

Cf. A065473, A126775, A196524, A256391, A330523, A336643.

Do[Print[N[Exp[-Sum[q = Expand[(4 p^2 - 4 p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}] (* Vaclav Kotesovec, Dec 17 2019 *)

prodeulerrat(1-4/p^2+4/p^3-1/p^4) \\ Amiram Eldar, Mar 03 2021

A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

Original entry on oeis.org

1, 4, 9, 4, 25, 36, 49, 4, 9, 100, 121, 36, 169, 196, 225, 4, 289, 36, 361, 100, 441, 484, 529, 36, 25, 676, 9, 196, 841, 900, 961, 4, 1089, 1156, 1225, 36, 1369, 1444, 1521, 100, 1681, 1764, 1849, 484, 225, 2116, 2209, 36, 49, 100, 2601, 676, 2809, 36, 3025, 196

Offset: 1

Reinhard Zumkeller, Dec 10 2002

It is conjectured that only 1 and 1782 satisfy a(k) = sigma(k). See Broughan link. - Michel Marcus, Feb 28 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.
K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.
Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.

Cf. A002117, A007948, A055231, A062378, A330523, A007434, A008683, A001221, A065958.

a := n -> mul(f,f=map(x->x^2,select(isprime,divisors(n))));
seq(a(n), n=1..56);  # Peter Luschny, Mar 30 2014

a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* Jean-François Alcover, Jun 04 2019 *)

a(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])^2 \\ Charles R Greathouse IV, Aug 06 2013

Multiplicative with a(p^e) = p^2. - Mitch Harris, May 17 2005
G.f.: Sum_{k>=1} mu(k)^2*J_2(k)*x^k/(1 - x^k), where J_2() is the Jordan function. - Ilya Gutkovskiy, Nov 06 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 / 3 = 0.214725... . - Amiram Eldar, Oct 30 2022
a(n) = Sum_{1 <= i, j <= n}  ( mobius(n/gcd(i, j, n)) )^2. - Peter Bala, Jan 28 2024
a(n) = Sum_{d|n} mu(d)^2*J_2(d), where J_2 = A007434. - Ridouane Oudra, Jul 24 2025
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*Psi_2(d), where omega = A001221 and Psi_2 = A065958. - Ridouane Oudra, Aug 01 2025

A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 11, 11, 18, 12, 24, 14, 24, 24, 20, 18, 33, 20, 36, 32, 36, 24, 44, 27, 42, 30, 48, 30, 72, 32, 37, 48, 54, 48, 66, 38, 60, 56, 66, 42, 96, 44, 72, 66, 72, 48, 80, 51, 81, 72, 84, 54, 90, 72, 88, 80, 90, 60, 144, 62, 96, 88, 70, 84, 144, 68

Offset: 1

Amiram Eldar, Feb 10 2023

a(n) is the sum of delta_d(n) over the divisors d of n, where delta_d(n) is the greatest divisor of n that is relatively prime to n.
Denoted by Sur(n) in Khan (2005).
Related sequences: A048691(n) = Sum_{d|n} #{d'; d' | n, gcd(d, d') = 1}, and A328485(n) = Sum_{d|n} Sum_{d' | n, gcd(d, d') = 1} d' (number and sum of divisors instead of maximal divisor, respectively).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mizan R. Khan, Problem 10922, The American Mathematical Monthly, Vol. 109, No. 2 (2002), p. 201; Michael R. Avidon, The Sum of Divisors Won't Die, solution, ibid., Vol. 110, No. 10 (2003), p. 959.
Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Cf. A000203, A005117, A034448, A360523.
Cf. A065465, A072691, A152649, A330523.
Cf. A048691, A328485.

f[p_, e_] := p^e + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]);}

Multiplicative with a(p^e) = p^e + e.
Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 * A065465 = A152649 * A330523 = 0.7250160726810604158... .
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
limsup_{n->oo} sigma(n)/a(n) = oo, where sigma(n) is the sum of divisors of n (A000203) (Khan, 2002).
liminf_{n->oo} a(n)/usigma(n) = 1, where usigma(n) is the sum of unitary divisors of n (A034448) (Khan, 2005).
limsup_{n->oo} a(n)/usigma(n) = (55/54) * Product_{p prime} (1 + 1/(p^2+1)) = 1.4682298236... (Khan, 2005).

A379715 The second Jordan totient function applied to the squarefree numbers.

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 840, 576, 960, 960, 864, 1152, 1368, 1080, 1344, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2880, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224, 3456, 5040, 5328, 4104, 5760, 4032

Offset: 1

Amiram Eldar, Dec 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mohammadreza Esfandiari, On the Means of Jordan's Totient Function, Bull. Iran. Math. Soc., Vol. 46 (2020), pp. 1753-1765.
R. Sitaramachandrarao, On an error term of Landau - II, Rocky Mountain J. Math., Vol. 15, No. 2 (1985), pp. 579-588. See p. 581.

Cf. A005117, A007434, A013661, A049200 (analogous with J_1 = phi), A330523, A379716, A379717, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; j2 /@ Select[Range[100], SquareFreeQ]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
list(lim) = apply(j2, select(issquarefree, vector(lim, i, i)));

a(n) = A007434(A005117(n)).
Sum_{n>=1} 1/a(n) = zeta(2) (A013661) (Sitaramachandrarao, 1985).
In general, Sum_{m squarefree} 1/J_k(m) = zeta(k), for k >= 2, where J_k is the k-th Jordan totient function.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2)^3 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) =  A013661^3 * A330523 = 2.38520727393117206135... . - Amiram Eldar, Jan 03 2025

A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Oct 21 2023

First differs from A274034 at n = 42, and from A197680 and A361177 at n = 84.

The asymptotic density of this sequence is zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 = A253905 * A065465 = 0.644177671086029533405... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Similar sequences with exponents of a given form: A000290 (2*k), A268335 (2*k+1), A000578 (3*k), A182120 (3*k+2).
Cf. A002117, A065465, A253905, A330523.
Cf. A197680, A274034, A361177, A366761.

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 1 &]; Select[Range[120], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 1, return(0))); 1;}

Sum_{n>=1} 1/a(n)^s = zeta(3*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)), for s > 1.

A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 14, 15, 13, 17, 16, 19, 20, 21, 22, 23, 21, 22, 26, 22, 28, 29, 30, 31, 24, 33, 34, 35, 32, 37, 38, 39, 35, 41, 42, 43, 44, 40, 46, 47, 39, 44, 44, 51, 52, 53, 44, 55, 49, 57, 58, 59, 60, 61, 62, 56, 46, 65, 66, 67, 68, 69, 70

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000010, A001616, A010052, A046790 (numbers n such that a(n) < n), A055653, A061884, A078779 (fixed points), A332619, A332686, A332712.

Table[Sum[EulerPhi[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 70}]
A055653[n_] := Sum[Boole[GCD[d, n/d] == 1] EulerPhi[d], {d, Divisors[n]}]; a[n_] := Sum[Boole[IntegerQ[(n/d)^(1/2)]] A055653[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

a(n) = sumdiv(n, d, eulerphi(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(s - 1) * Product_{p prime} (1 - p^(-s) + p^(-2*s) - p^(1 - 2*s)).
a(n) = Sum_{d|n} phi(lcm(d, n/d)/d).
a(n) = Sum_{d|n} A010052(n/d) * A055653(d).
Sum_{k=1..n} a(k) ~ c * Pi^6 * n^2 / 1080, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Feb 22 2020
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(n/gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))*gcd(n,k)/n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(gcd(n,k))*A055653(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(n/gcd(n,k))*A055653(gcd(n,k))/phi(n/gcd(n,k)). (End)

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A062354 a(n) = sigma(n)*phi(n).

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A000252 Number of invertible 2 X 2 matrices mod n.

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A055653 Sum of phi(d) [A000010] over all unitary divisors d of n (that is, gcd(d,n/d) = 1).

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A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

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A256392 Decimal expansion of Product_{p prime} (1-4/p^2+4/p^3-1/p^4).

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A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

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A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

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