A065464 -id:A065464 - cp's OEIS frontend

A078078 Continued fraction for constant defined in A065464.

0, 2, 2, 1, 62, 1, 2, 7, 1, 4, 1, 3, 5, 1, 14, 5, 1, 2, 3, 5, 7, 1, 2, 3, 3, 5, 1, 13, 7, 2, 1, 5, 1, 1, 1, 2, 7, 1, 3, 1, 1, 1, 2, 2, 4, 7, 1, 4, 6, 2, 1, 2, 1, 12, 3, 2, 7, 4, 1, 2, 3, 18, 5, 1, 2, 5, 1, 3, 35, 1, 2, 1, 22, 2, 3, 23, 3, 5, 83, 2, 1, 1, 2, 2, 2, 1, 22, 2, 56, 3, 1, 7, 1, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065464 (decimal expansion).

contfrac(prodeulerrat(1 - (2*p-1)/p^3)) \\ Amiram Eldar, Mar 12 2021

Offset changed by Andrew Howroyd, Jul 05 2024

A047994 Unitary totient (or unitary phi) function uphi(n).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 7, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 14, 24, 12, 26, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 28, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 26, 40, 42, 36, 28, 58, 24, 60, 30, 48, 63, 48, 20, 66, 48, 44, 24, 70

Offset: 1

N. J. A. Sloane

A divisor d of n is called a unitary divisor if gcd(d, n/d) = 1. Define gcd*(k,n) to be the largest divisor d of k that is also a unitary divisor of n (that is, such that gcd(d, n/d) = 1). The unitary totient function a(n) = number of k with 1 <= k <= n such that gcd*(k,n) = 1. - N. J. A. Sloane, Aug 08 2021
Unitary convolution of A076479 and A000027. - R. J. Mathar, Apr 13 2011
Multiplicative with a(p^e) = p^e - 1. - N. J. A. Sloane, Apr 30 2013

			a(12) = a(3)*a(4) = 2*3 = 6.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80
Steven R. Finch, Unitarism and infinitarism.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
M. Lal, Iterates of the unitary totient function, Math. Comp., 28 (1974), 301-302.
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011, Remark 43.
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2.
László Tóth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010), Article 10.8.1.

Cf. A000010, A000203, A001221, A003557, A049865, A003271, A028233, A076479.

a047994 n = f n 1 where
   f 1 uph = uph
   f x uph = f (x `div` sppf) (uph * (sppf - 1)) where sppf = a028233 x
-- Reinhard Zumkeller, Aug 17 2011

A047994 := proc(n)
    local a, f;
    a := 1 ;
    for f in ifactors(n)[2] do
        a := a*(op(1,f)^op(2,f)-1) ;
    end do:
    a ;
end proc:
seq(A047994(n),n=1..20) ; # R. J. Mathar, Dec 22 2011

uphi[n_] := (Times @@ (Table[ #[[1]]^ #[[2]] - 1, {1} ] & /@ FactorInteger[n]))[[1]]; Table[ uphi[n], {n, 2, 75}] (* Robert G. Wilson v, Sep 06 2004 *)
uphi[n_] := If[n==1, 1, Product[{p, e} = pe; p^e-1, {pe, FactorInteger[n]}] ]; Array[uphi, 80] (* Jean-François Alcover, Nov 17 2018 *)

A047994(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

from math import prod
from sympy import factorint
def A047994(n): return prod(p**e-1 for p, e in factorint(n).items()) # Chai Wah Wu, Sep 24 2021

If n = Product p_i^e_i, uphi(n) = Product (p_i^e_i - 1).
a(n) = A000010(n)*A000203(A003557(n))/A003557(n). - Velin Yanev and Charles R Greathouse IV, Aug 23 2017
From Amiram Eldar, May 29 2020: (Start)
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(d) * n/d.
Sum_{d|n, gcd(d, n/d) = 1} a(d) = n.
a(n) >= phi(n) = A000010(n), with equality if and only if n is squarefree (A005117). (End)
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3). - Vaclav Kotesovec, Jun 15 2020
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)). - Amiram Eldar, May 22 2025

More terms from Jud McCranie

A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 2, 7, 1, 8, 1, 9, 8, 4, 1, 7, 1, 12, 10, 13, 1, 11, 2, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 10, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 2, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 6, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 4, 43, 1, 62, 22, 45, 32, 35, 1, 41, 20

Offset: 1

Antti Karttunen, Feb 28 2021

nonn

See also the scatter plot of A342002 that seems to reveal some interesting internal structure in this sequence, not fully explained by the regularity of primorial base expansion used in the latter sequence. - Antti Karttunen, May 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A341998, A342416, A342459.
Cf. A342002 [= a(A276086(n))], A342463 [= a(A342456(n))], A351945 [= a(A181819(n))], A353571 [= a(A003961(n))].
Cf. A346485 (Möbius transform), A347395 (convolution with Liouville's lambda), A347961 (with itself), and A347234, A347235, A347954, A347959, A347963, A349396, A349612 (for convolutions with other sequences).
Cf. A007947.

Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));

from math import prod
from sympy import factorint
def A342001(n):
    q = prod(f:=factorint(n))
    return sum(q*e//p for p, e in f.items()) # Chai Wah Wu, Nov 04 2022

a(n) = A003415(n) / A003557(n).
For all n >= 0, a(A276086(n)) = A342002(n).
a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - Antti Karttunen, Apr 30 2022
Dirichlet g.f.: Dirichlet g.f. of A007947 * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)) = zeta(s) * Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022
Sum_{k=1..n} a(k) ~ c * A065464 * Pi^2 * n^2 / 12, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, May 09 2022

A064549 a(n) = n * Product_{primes p|n} p.

Original entry on oeis.org

1, 4, 9, 8, 25, 36, 49, 16, 27, 100, 121, 72, 169, 196, 225, 32, 289, 108, 361, 200, 441, 484, 529, 144, 125, 676, 81, 392, 841, 900, 961, 64, 1089, 1156, 1225, 216, 1369, 1444, 1521, 400, 1681, 1764, 1849, 968, 675, 2116, 2209, 288, 343, 500, 2601, 1352

Offset: 1

Henry Bottomley, Oct 16 2001

Index of first occurrence of n in A003557. - Franklin T. Adams-Watters, Jul 25 2014

			a(12) = 72 since 12 = 2^2*3 and 12*2*3 = 72.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, Vol. 113, No. 8 (2006), pp. 1732-1745; arXiv preprint, arXiv:math/0509316 [math.NT], 2005-2006.

A permutation of the powerful numbers A001694.

Cf. A003557 (a left inverse), A007947, A057521, A078310, A082695, A202535.

a064549 n = a007947 n * n  -- Reinhard Zumkeller, Jul 23 2013

[n^2/( (&+[Floor(k^n/n)-Floor((k^n - 1)/n) : k in [1..n]]) ): n in [1..50]]; // G. C. Greubel, Nov 02 2018

a:= n -> n * convert(numtheory:-factorset(n), `*`):
seq(a(n),n=1..100); # Robert Israel, Jul 25 2014

a[n_] := n * Times @@ FactorInteger[n][[All, 1]]; Array[a, 100] (* Jean-François Alcover, Feb 17 2017 *)
Table[n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jun 15 2019 *)

popf(n)= { local(f,p=1); f=factor(n); for(i=1, matsize(f)[1], p*=f[i, 1]); return(p) } { for (n=1, 1000, write("b064549.txt", n, " ", n*popf(n)) ) } \\ Harry J. Smith, Sep 18 2009

A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); }; \\ Antti Karttunen, Aug 30 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

Multiplicative with a(p^k)=p^(k+1) when k>0.
a(n) = n*A007947(n) = n^2/A003557(n).
Dirichlet convolution of A000027 and A202535. - R. J. Mathar, Dec 20 2011
a(n) = A078310(n) - 1. - Reinhard Zumkeller, Jul 23 2013
A003557(a(n)) = n; a(A003557(n)) = A057521(n). - Antti Karttunen, Aug 30 2018
G.f.: Sum_{k>=1} mu(k)^2*phi(k)*k*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
From Vaclav Kotesovec, Jun 24 2020: (Start)
Dirichlet g.f.: zeta(s-2) * zeta(s-1) * Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A065463/3 = A065464*Pi^2/18 = 0.234814...
(End)
Sum_{k>=1} 1/a(k) = zeta(2)*zeta(3)/zeta(6) = A082695. - Vaclav Kotesovec, Sep 19 2020
Sum_{k>=1} (-1)^(k+1)/a(k) = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 18 2020

A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 16, 18, 16, 20, 24, 24, 24, 32, 40, 32, 36, 36, 48, 48, 40, 44, 64, 60, 48, 72, 72, 56, 64, 60, 96, 80, 64, 96, 108, 72, 72, 96, 128, 80, 96, 84, 120, 144, 88, 92, 160, 126, 120, 128, 144, 104, 144, 160, 192, 144, 112, 116, 192, 120, 120, 216, 224

Offset: 1

Jason Earls, Jul 06 2001

a(n) = sum of gcd(k-1,n) for 1 <= k <= n and gcd(k,n)=1 (Menon's identity).
For n = 2^(4*k^2 - 1), k >= 1, the terms of the sequence are square and for n = 2^((3*k + 2)^3 - 1), k >= 1, the terms of the sequence are cubes. - Marius A. Burtea, Nov 14 2019
Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 20 2020

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.
P. K. Menon, On the sum gcd(a-1,n) [(a,n)=1], J. Indian Math. Soc. (N.S.), 29 (1965), 155-163.
József Sándor, On Dedekind's arithmetical function, Seminarul de teoria structurilor (in Romanian), No. 51, Univ. Timișoara, 1988, pp. 1-15. See p. 11.
József Sándor, Some diophantine equations for particular arithmetic functions (in Romanian), Seminarul de teoria structurilor, No. 53, Univ. Timișoara, 1989, pp. 1-10. See p. 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Vaclav Kotesovec, Graph - the asymptotic ratio (250000000 terms)
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, Section 3.15.
R. Sivaramakrishnan, Problem E 1962, Elementary Problems, The American Mathematical Monthly, Vol. 74, No. 2 (1967), p. 198; Solution, ibid., Vol. 75, No. 5 (1968), p. 550.
Marius Tarnauceanu, A generalization of the Menon's identity, arXiv:1109.2198 [math.GR], 2011-2012.
Laszlo Toth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
Wikipedia, Arithmetic function (Menon's identity).

Cf. A003557, A173557, A061468, A062816, A079535, A062949 (inverse Mobius transform), A304408, A318519, A327169 (number of times n occurs in this sequence).

Cf. A062354, A064840.

[NumberOfDivisors(n)*EulerPhi(n):n in [1..65]]; // Marius A. Burtea, Nov 14 2019

seq(tau(n)*phi(n), n=1..64); # Zerinvary Lajos, Jan 22 2007

Table[EulerPhi[n] DivisorSigma[0, n], {n, 80}] (* Carl Najafi, Aug 16 2011 *)
f[p_, e_] := (e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

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

{ for (n=1, 1000, write("b062355.txt", n, " ", numdiv(n)*eulerphi(n)) ) } \\ Harry J. Smith, Aug 05 2009

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

Dirichlet convolution of A047994 and A000010. - R. J. Mathar, Apr 15 2011
a(n) = A000005(n)*A000010(n). Multiplicative with a(p^e) = (e+1)*(p-1)*p^(e-1). - R. J. Mathar, Jun 23 2018
a(n) = A173557(n) * A318519(n) = A003557(n) * A304408(n). - Antti Karttunen, Sep 16 2018 & Sep 20 2019
From Vaclav Kotesovec, Jun 15 2020: (Start)
Let f(s) = Product_{primes p} (1 - 2*p^(-s) + p^(1-2*s)).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ n^2 * (f(2)*(log(n)/2 + gamma - 1/4) + f'(2)/2), where f(2) = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.42824950567709444...,
f'(2) = 2 * A065464 * A335707 = f(2) * Sum_{primes p} 2*log(p) / (p^2 + p - 1) = 0.35866545223424232469545420783620795... and gamma is the Euler-Mascheroni constant A001620. (End)
From Amiram Eldar, Mar 02 2021: (Start)
a(n) >= n (Sivaramakrishnan, 1967).
a(n) >= sigma(n), for odd n (Sándor, 1988).
a(n) >= phi(n) + n - 1 (Sándor, 1989) (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} uphi(gcd(n,k)), where uphi(n) = A047994(n).
a(n) = Sum_{k=1..n} uphi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A127473 a(n) = phi(n)^2.

Original entry on oeis.org

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 64, 64, 256, 36, 324, 64, 144, 100, 484, 64, 400, 144, 324, 144, 784, 64, 900, 256, 400, 256, 576, 144, 1296, 324, 576, 256, 1600, 144, 1764, 400, 576, 484, 2116, 256, 1764, 400, 1024, 576, 2704, 324, 1600

Offset: 1

Gary W. Adamson, Jan 15 2007

Number of maps of the form j |--> m*j + d with gcd(m, n) = 1 and gcd(d, n) = 1 from [1, 2, ..., n] to itself. - Joerg Arndt, Aug 29 2014
Right border of A127474.
Equals the Mobius transform (A054525) of A029939. - Gary W. Adamson, Aug 20 2008
From Jianing Song, Apr 14 2019: (Start)
a(n) is the number of solutions to gcd(xy, n) = 1 with x, y in [0, n-1].
Let Z_n be the ring of integers modulo n, then a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - x) (or equivalently, Z_n[x]/(x^2 + x)) with discriminant d = 1 (that is, a(n) is the size of the group G(n) = (Z_n[x]/(x^2 - x))*). Actually, G(n) is isomorphic to (Z_n)* X (Z_n)*. (End)

			a(5) = 16 since phi(5) = 4.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Cf. A000010, A057434, A065464, A109695, A127474, A358714, A361148, A008683.

Similar sequences: A082953 (size of (Z_n[x]/(x^2 - 1))*, d = 4), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

[(EulerPhi(n))^2: n in [1..180]]; // Vincenzo Librandi, Apr 04 2011

A127473 := proc(n) numtheory[phi](n)^2 ; end proc:
seq(A127473(n),n=1..40) ; # R. J. Mathar, Apr 04 2011

Table[EulerPhi[n]^2,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

a(n) = eulerphi(n)^2; \\ Michel Marcus, Oct 16 2018

a(n) = A000010(n)^2.
Multiplicative with a(p^e) = (p-1)^2*p^(2e-2), e >= 1. Dirichlet g.f. zeta(s-2)*Product_{primes p} (1 - 2/p^(s-1) + 1/p^s). - R. J. Mathar, Apr 04 2011
Sum_{k>=1} 1/a(k) = A109695. - Vaclav Kotesovec, Sep 20 2020
Sum_{k>=1} (-1)^k/a(k) = (1/7) * A109695. - Amiram Eldar, Nov 11 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime}(1 - (2*p-1)/p^3) = A065464 / 3 = 0.142749... . - Amiram Eldar, Oct 25 2022
a(n) = Sum_{d|n} mu(n/d)*phi(n*d). - Ridouane Oudra, Jul 23 2025

A097945 a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010).

Original entry on oeis.org

1, -1, -2, 0, -4, 2, -6, 0, 0, 4, -10, 0, -12, 6, 8, 0, -16, 0, -18, 0, 12, 10, -22, 0, 0, 12, 0, 0, -28, -8, -30, 0, 20, 16, 24, 0, -36, 18, 24, 0, -40, -12, -42, 0, 0, 22, -46, 0, 0, 0, 32, 0, -52, 0, 40, 0, 36, 28, -58, 0, -60, 30, 0, 0, 48, -20, -66, 0, 44, -24, -70, 0, -72, 36, 0, 0, 60, -24, -78, 0, 0, 40, -82, 0

Offset: 1

Gerald McGarvey, Sep 04 2004

Also, a(n) = mu(n)*uphi(n) where mu(n) is the Mobius function (A008683) and uphi(n) is the unitary totient function (A047994), since phi(n) = uphi(n) when n is squarefree, while mu(n) = 0 when n is not squarefree. - Franklin T. Adams-Watters, May 14 2006
Conjecture: Sum_{n>=1} mu(n)/phi(n) = Sum_{n>=1} a(n)/phi(n)^2 = 0. It is true that Sum_{n>=1} mu(n)/phi(n)^s = 0 at least for s > 1 since: phi(2)=1, phi is multiplicative, so for n's that are squarefree, the phi(n) values can be partitioned in pairs where phi(m)=phi(2m) and mu(m) = -mu(2m). So Sum_{i=1..n} mu(i)/phi(i)^s < Sum_{j=floor(n/2)..n} 1/phi(j)^s, which approaches 0 as n increases since (1) n^(1-e) < phi(n) < n for any e > 0 and n > N(e) and (2) Sum_{i..n} 1/n^s converges for s > 1. Conjecture: Sum_{n>=1} mu(n)/phi(n)^z = 0 for Re(z) > 1.
Multiplicative with a(p^1) = 1-p, a(p^e) = 0, e > 1. - Mitch Harris, May 24 2005
Row sums of triangle A143153 = a signed version of the sequence such that parity = (-) iff A008683(n) = (+); 0 or (+): (1, 1, 2, 0, 4, -2, 6, 0, 0, -4, 10, 0, 12, -6, 0, 0, 0, ...). - Gary W. Adamson, Jul 27 2008
Dirichlet inverse of A003958. - R. J. Mathar, Jul 08 2011

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010, A003958, A007947, A008683, A008966, A023900, A047994, A143153, A173557, A322581.

with(numtheory):
a:= n-> mobius(n)*phi(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 06 2012

Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (* Robert G. Wilson v, Sep 06 2004 *)

a(n)=moebius(n)*eulerphi(n) \\ Charles R Greathouse IV, Feb 21 2013

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Dirichlet g.f.: Product_{primes p} (1-p^(1-s)+p^(-s)). - R. J. Mathar, Aug 29 2011
Sum_{d|n} abs(a(d)) = rad(n) = A007947(n). - Rémy Sigrist, Nov 05 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = A065464/2 = (1/2) * Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.21412475283854722... Equivalently, c = A065463 * 3 / Pi^2. - Vaclav Kotesovec, Jun 14 2020
From Antti Karttunen, Aug 20 2021: (Start)
a(n) = mu(n)*A000010(n) = mu(n)*A003958(n) = mu(n)*A047994(n) = mu(n)*A173557(n), where mu is Möbius mu function (A008683).
a(n) = A008966(n) * A023900(n) = abs(mu(n)) * A023900(n).
a(n) = A322581(n) - A003958(n).
(End)

More terms from Robert G. Wilson v, Sep 06 2004

Edited by N. J. A. Sloane, May 20 2006

A322483 The number of semi-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Dec 11 2018

The notion of semi-unitary divisor was introduced by Chidambaraswamy in 1967.
A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1. In terms of the relation defined in A322482, d is the largest divisor of n such that T(d, n/d) = 1 (the largest divisor d that is semiprime to n/d).
The number of divisors of n that are exponentially odd numbers (A268335). - Amiram Eldar, Sep 08 2023

			The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their number is 3, thus a(8) = 3.

J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, Journal of the Australian Mathematical Society, Vol. 23, No. 1 (1977), pp. 9-27.
Pentti Haukkanen, Basic properties of the bi-unitary convolution and the semi-unitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305-315.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions. Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of unitarily k-free divisors of an integer, Journal of the Australian Mathematical Society, Vol. 21, No. 1 (1976), pp. 19-35.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.

Cf. A000005, A005117, A019554, A034444, A268335, A286324, A322482.

Cf. A001620, A056671, A343443, A365488, A365491.

f[p_, e_] := Floor[(e+3)/2]; sud[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[sud, 100]

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = (f[k,2]+3)\2; f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Dec 14 2018

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1-X^2) * (1 + X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 06 2023

Multiplicative with a(p^e) = floor((e+3)/2).
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
a(n) = Sum_{d|n} mu(d/gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
a(n) = A000005(A019554(n)) (the number of divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 02 2023
From Vaclav Kotesovec, Sep 06 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Let f(s) = Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444...,
f'(1) = f(1) * Sum_{p prime} (4*p-3) * log(p) / (p^3 - 2*p + 1) = 0.808661108949590913395... and gamma is the Euler-Mascheroni constant A001620. (End)

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 19 2019

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A056671 1 + the number of unitary and squarefree divisors of n = number of divisors of reduced squarefree part of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 1, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 4, 1, 2, 2, 8, 2, 1, 4, 4, 4, 1, 2, 4, 4, 2, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 2, 4, 2, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 1, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 1, 2, 8, 2, 2, 8

Offset: 1

Labos Elemer, Aug 10 2000

Note that 1 is regarded as free of squares of primes and is also a square number and a unitary divisor.

			n = 252 = 2*2*3*3*7 has 18 divisors, 8 unitary and 8 squarefree divisors of which 2 are unitary and squarefree, divisors {1,7};
n = 2520 = 2*2*2*3*3*5*7 has 48 divisors, 16 unitary and 16 squarefree divisors of which {1,5,7,35} are both, thus a(2520) = 4.
a(2520) = a(2^3*3^2*5*7) = a(2^3)*a(3^2)*a(5)*a(7) = 1*1*2*2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A007913, A034444, A055229, A055231, A056169, A056674.

Cf. A343443.

Array[DivisorSigma[0, #] &@ Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] &, 105] (* or *)
Table[DivisorSum[n, 1 &, And[SquareFreeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)
f[p_,e_] := If[e==1, 2, 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ Charles R Greathouse IV, Aug 13 2013
A055231(n) = n/A057521(n);
A056671(n) = numdiv(A055231(n));
\\ Or:
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ This function from Charles R Greathouse IV, Nov 20 2012
A056671(n) = numdiv(core(n)/A055229(n)); \\ Antti Karttunen, Jul 19 2017

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

a(n) = vecprod(apply(x -> if(x == 1, 2, 1), factor(n)[, 2])); \\ Amiram Eldar, Apr 15 2025

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod([2 if e==1 else 1 for p, e in factorint(n).items()])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

(define (A056671 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) 2 1) (A056671 (A028234 n))))) ;; (After the given multiplicative formula) - Antti Karttunen, Jul 19 2017

a(n) = A000005(A055231(n)) = A000005(A007913(n)/A055229(n)).
Multiplicative with a(p) = 2 and a(p^e) = 1 for e > 1. a(n) = 2^A056169(n). - Vladeta Jovovic, Nov 01 2001
a(n) = A034444(n) - A056674(n). - Antti Karttunen, Jul 19 2017
From Vaclav Kotesovec, Feb 11 2023: (Start)
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 2/p^(2*s) + 1/p^(3*s)), (with a product that converges for s=1).
Let f(s) = Product_{primes p} (1 - 2/p^(2*s) + 1/p^(3*s)), then Sum_{k=1..n} a(k) ~ n * (f(1) * (log(n) + 2*gamma - 1) + f'(1)), where f(1) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444021876..., f'(1) = f(1) * Sum_{primes p} (4*p-3) * log(p) / (p^3 - 2*p + 1) = 0.808661108949590913395... and gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = Sum_{d|n, gcd(d,n/d)=1} mu(d)^2. - Wesley Ivan Hurt, May 25 2023
a(n) = Sum_{d|n} A343443(d)*mu(n/d). - Ridouane Oudra, Dec 18 2023

cp's OEIS Frontend

A078078 Continued fraction for constant defined in A065464.

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A047994 Unitary totient (or unitary phi) function uphi(n).

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A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

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A064549 a(n) = n * Product_{primes p|n} p.

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A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

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A127473 a(n) = phi(n)^2.

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