A007431 -id:A007431 - cp's OEIS frontend

A130054 Inverse Moebius transform of A023900.

1, 0, -1, -1, -3, 0, -5, -2, -3, 0, -9, 1, -11, 0, 3, -3, -15, 0, -17, 3, 5, 0, -21, 2, -7, 0, -5, 5, -27, 0, -29, -4, 9, 0, 15, 3, -35, 0, 11, 6, -39, 0, -41, 9, 9, 0, -45, 3, -11, 0, 15, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21

Offset: 1

Gary W. Adamson, May 04 2007

Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Row sums of A129691.

Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

[&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019

with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019

b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)

\\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018

A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018
a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019
From Werner Schulte, Sep 06 2020: (Start)
Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
Dirichlet convolution with A062570 equals A001511.
Dirichlet convolution with A018804 equals A000203.
Dirichlet inverse of A007431. (End)
a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A062790 Moebius transform of the cototient function A051953.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 3, 1, 6, 5, 4, 1, 6, 1, 5, 7, 10, 1, 6, 4, 12, 6, 7, 1, 8, 1, 8, 11, 16, 9, 8, 1, 18, 13, 10, 1, 12, 1, 11, 12, 22, 1, 12, 6, 20, 17, 13, 1, 18, 13, 14, 19, 28, 1, 13, 1, 30, 16, 16, 15, 20, 1, 17, 23, 24, 1, 16, 1, 36, 24, 19, 15, 24, 1, 20, 18, 40, 1, 19

Offset: 1

Labos Elemer, Jul 19 2001

nonn

			n = 255, its divisors are {1,3,5,25,17,51,85,255}, A051953(255/d) = {127,21,19,1,7,1,1,0}, mu(d) = {1,-1,-1,1,-1,1,1,-1}, the sum is a(255) = 127-21-19+1-7+1+1+0 = 130-47 = 83.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1 .. 2000 from Harry J. Smith)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A000010, A001065, A007431, A051953, A056239, A318836.

Table[DirichletConvolve[MoebiusMu[n], n-EulerPhi[n], n, k], {k, 100}]  (* Amiram Eldar, Nov 24 2018 *)

A062790(n)={
    local(a=0) ;
    fordiv(n,d,
        a += moebius(d)*(n/d-eulerphi(n/d)) ;
    ) ;
    return(a) ;
} \\ R. J. Mathar, Mar 24 2012

A062790(n) = sumdiv(n,d,moebius(n/d)*(d-eulerphi(d))); \\ Antti Karttunen, Nov 24 2018

a(n) = Sum f(n/d)*mu(d), where d divides n and f(x) = x-phi(x) = A051953(x).
a(n) = A056239(A318836(n)). - Antti Karttunen, Nov 24 2018
From Amiram Eldar, Dec 15 2023: (Start)
a(n) = A000010(n) - A007431(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 6/Pi^2 - 36/Pi^4. (End)

OFFSET changed from 0 to 1 by Harry J. Smith, Aug 11 2009

A354058 Square array read by ascending antidiagonals: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 5, 0, 3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1

Offset: 1

Jianing Song, May 16 2022

Given n, T(n,k) only depends on gcd(k,psi(n)). For the truncated version see A354061.
Each column is multiplicative.
The n-th rows contains entirely 0's if and only if n == 2 (mod 4).
For n !== 2 (mod 4), T(n,psi(n)) > T(n,k) if k is not divisible by psi(n).
Proof: this is true if n is a prime power (see the formula below). Now suppose that n = Product_{i=1..r} (p_i)^(e_i). Since n !== 2 (mod 4), (p_i)^(e_i) != 2, so T((p_i)^(e_i),psi((p_i)^(e_i))) > 0 for each i. If k is not divisible by psi(n), then it is not divisible by some psi((p_{i_0})^(e_{i_0})), so T(n,psi(n)) = Product_{i=1..r} T((p_i)^(e_i),psi(n)) = Product_{i=1..r} T((p_i)^(e_i),psi((p_i)^(e_i))) > T((p_{i_0})^(e_{i_0}),k) * Product_{i!=i_0} T((p_i)^(e_i),psi((p_i)^(e_i))) >= Product_{i=1..r} T((p_i)^(e_i),k) = T(n,k).

			  n/k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
   1   1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
   2   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   3   0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
   4   0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
   5   0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3
   6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   7   0  1  2  1  0  5  0  1  2  1  0  5  0  1  2  1  0  5  0  1
   8   0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2
   9   0  0  2  0  0  4  0  0  2  0  0  4  0  0  2  0  0  4  0  0
  10   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  11   0  1  0  1  4  1  0  1  0  9  0  1  0  1  4  1  0  1  0  9
  12   0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
  13   0  1  2  3  0  5  0  3  2  1  0 11  0  1  2  3  0  5  0  3
  14   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  15   0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3
  16   0  0  0  4  0  0  0  4  0  0  0  4  0  0  0  4  0  0  0  4
  17   0  1  0  3  0  1  0  7  0  1  0  3  0  1  0 15  0  1  0  3
  18   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  19   0  1  2  1  0  5  0  1  8  1  0  5  0  1  2  1  0 17  0  1
  20   0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3  0  1  0  3

Jianing Song, Table of n, a(n) for n = 1..5050 (the first 100 ascending diagonals)

k-th column: A114643 (k=2), A160498 (k=3), A160499 (k=4), A307380 (k=5), A307381 (k=6), A307382 (k=7), A329272 (k=8).
Moebius transform of A354057 applied to each column.
A354257 gives the smallest index for the nonzero terms in each row.
Cf. A007431.

b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n,k) = sumdiv(n, d, moebius(n/d)*b(d,k))

For odd primes p: T(p,k) = gcd(p-1,k)-1, T(p^e,k*p^(e-1)) = p^(e-2)*(p-1)*gcd(k,p-1), T(p^e,k) = 0 if k is not divisible by p^(e-1). T(2,k) = 0, T(4,k) = 1 for even k and 0 for odd k, T(2^e,k) = 2^(e-2) if k is divisible by 2^(e-2) and 0 otherwise.

T(n,psi(n)) = A007431(n). - Jianing Song, May 24 2022

A354061 Irregular table read by rows: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n, 1 <= k <= psi(n), psi = A002322.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 2, 1, 0, 5, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 1, 4, 1, 0, 1, 0, 9, 0, 1, 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15

Offset: 1

Jianing Song, May 16 2022

Given n, T(n,k) only depends on gcd(k,psi(n)).
The n-th row contains entirely 0's if and only if n == 2 (mod 4).
If n !== 2 (mod 4), T(n,psi(n)) > T(n,k) for 1 <= k < psi(n).

			Table starts
n = 1: 1;
n = 2: 0;
n = 3: 0, 1;
n = 4: 0, 1;
n = 5: 0, 1, 0, 3;
n = 6: 0, 0;
n = 7: 0, 1, 2, 1, 0, 5;
n = 8: 0, 2;
n = 9: 0, 0, 2, 0, 0, 4;
n = 10: 0, 0, 0, 0;
n = 11: 0, 1, 0, 1, 4, 1, 0, 1, 0, 9;
n = 12: 0, 1;
n = 13: 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11;
n = 14: 0, 0, 0, 0, 0, 0;
n = 15: 0, 1, 0, 3;
n = 16: 0, 0, 0, 4;
n = 17: 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15;
n = 18: 0, 0, 0, 0, 0, 0;
n = 19: 0, 1, 2, 1, 0, 5, 0, 1, 8, 1, 0, 5, 0, 1, 2, 1, 0, 17;
n = 20: 0, 1, 0, 3;
...

Jianing Song, Table of n, a(n) for n = 1..8346 (the first 200 rows)

Cf. A354058, A002322, A007431, A354060.

A354257 gives the smallest index for the nonzero terms in each row.

b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n,k) = sumdiv(n, d, moebius(n/d)*b(d,k))

For odd primes p: T(p,k) = gcd(p-1,k)-1, T(p^e,k*p^(e-1)) = p^(e-2)*(p-1)*gcd(k,p-1), T(p^e,k) = 0 if k is not divisible by p^(e-1). T(2,k) = 0, T(4,k) = 1 for even k and 0 for odd k, T(2^e,k) = 2^(e-2) if k is divisible by 2^(e-2) and 0 otherwise.

T(n,psi(n)) = A007431(n). - Jianing Song, May 24 2022

A318839 Restricted growth sequence transform of A318838.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 5, 6, 3, 7, 8, 9, 4, 10, 11, 12, 6, 13, 10, 14, 7, 15, 16, 17, 9, 18, 14, 19, 10, 20, 21, 22, 12, 23, 24, 25, 13, 26, 27, 28, 14, 29, 22, 30, 15, 31, 32, 33, 17, 34, 26, 35, 18, 36, 37, 38, 19, 39, 40, 41, 20, 42, 43, 44, 22, 45, 34, 46, 23, 47, 48, 49, 25, 50, 38, 51, 26, 52, 53, 54, 28, 55, 56, 57, 29, 58, 59, 60, 30, 61, 46, 62, 31, 63

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

For all i, j: a(i) = a(j) => A000010(i) = A000010(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A007431, A318838.

Cf. also A318837.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007431(n) = sumdiv(n,d,moebius(n/d)*eulerphi(d));
A318838(n) = { my(m=1); fordiv(n,d,if((A007431(d)!=0),m *= prime(A007431(d)))); (m); };
v318839 = rgs_transform(vector(up_to,n,A318838(n)));
A318839(n) = v318839[n];

A321322 a(n) = Sum_{d|n} mu(n/d)*J_2(d), where J_2() is the Jordan function (A007434).

Original entry on oeis.org

1, 2, 7, 9, 23, 14, 47, 36, 64, 46, 119, 63, 167, 94, 161, 144, 287, 128, 359, 207, 329, 238, 527, 252, 576, 334, 576, 423, 839, 322, 959, 576, 833, 574, 1081, 576, 1367, 718, 1169, 828, 1679, 658, 1847, 1071, 1472, 1054, 2207, 1008, 2304, 1152, 2009, 1503, 2807, 1152, 2737

Offset: 1

Ilya Gutkovskiy, Nov 04 2018

Möbius transform applied twice to squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000010, A001615, A007427, A007431, A007433, A007434, A008683.

Table[Sum[MoebiusMu[n/d] Sum[MoebiusMu[d/j] j^2, {j, Divisors[d]}], {d, Divisors[n]}], {n, 55}]
nmax = 55; Rest[CoefficientList[Series[Sum[DivisorSum[k, MoebiusMu[#] MoebiusMu[k/#] &] x^k (1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
f[p_, e_] := If[e == 1, p^2 - 2, (p^2 - 1)^2*p^(2*e - 4)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^2/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Dec 11 2021

G.f.: Sum_{k>=1} A007427(k)*x^k*(1 + x^k)/(1 - x^k)^3.
a(n) = Sum_{d|n} mu(n/d)*phi(d)*psi(d), where phi() is the Euler totient function (A000010) and psi() is the Dedekind psi function (A001615).
Multiplicative with a(p^e) = p^2 - 2 if e = 1 and (p^2 - 1)^2 * p^(2*e - 4) otherwise. - Amiram Eldar, Oct 26 2020
From Vaclav Kotesovec, Dec 11 2021: (Start)
Dirichlet g.f.: zeta(s-2) / zeta(s)^2.
Sum_{k=1..n} a(k) ~ n^3 / (3*zeta(3)^2). (End)
a(n) = Sum_{1 <= i, j <= n} mu(gcd(i, j, n)). - Peter Bala, Jan 21 2024

A340190 Möbius transform of A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

Original entry on oeis.org

1, 0, 1, 0, 3, -1, 5, 0, 0, -3, 9, 0, 11, -5, -1, 0, 15, 0, 17, 0, -3, -9, 21, 0, 0, -11, 0, 2, 27, 1, 29, 0, -7, -15, -5, 0, 35, -17, -9, 0, 39, 3, 41, 0, 4, -21, 45, 0, 0, 0, -13, 2, 51, 0, -9, -2, -15, -27, 57, 0, 59, -29, 0, 0, 1, 11, 65, 0, -19, 7, 69, 0, 71, -35, 0, 2, -11, 9, 77, 0, 0, -39, 81, -2, -3, -41, -25

Offset: 1

Antti Karttunen, Dec 31 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008683, A063994, A340191.

Cf. also A007431, A340143, A340146, A340192.

A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
A340190(n) = sumdiv(n,d,moebius(n/d)*A063994(d));

a(n) = Sum_{d|n} A008683(n/d) * A063994(d).

a(n) = A063994(n) - A340191(n).

A114810 Number of complex, weakly primitive Dirichlet characters modulo n.

Original entry on oeis.org

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

Offset: 1

Steven Finch, Feb 19 2006

Any primitive Dirichlet character is weakly primitive (not conversely). Jager uses the phrase "proper character", but this conflicts with other authors (e.g., W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, p. 224) who use the word "proper" to mean the same as "primitive".

Equals Mobius transform of A055653. - Gary W. Adamson, Feb 28 2009

			The function chi defined on the integers by chi(1)=1, chi(5)=-1 and chi(2)=chi(3)=chi(4)=chi(6)=0 [and extended periodically] is a weakly primitive character mod 6, but not mod 12 or mod 18. In this sense, we eliminate the "overcounting" of complex Dirichlet characters in A000010.

Amiram Eldar, Table of n, a(n) for n = 1..10000
H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.

Cf. A000010, A007431, A055231, A330523.

Cf. A055653. [Gary W. Adamson, Feb 28 2009]

b[n_] := Sum[EulerPhi[d]*MoebiusMu[n/d], {d, Divisors[n]}]; squareFreeKernel[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Sum[b[n/d], {d, Divisors[Denominator[n/squareFreeKernel[n]^2]]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Sep 07 2015 *)
f[p_, e_] := If[e == 1, p - 1, (p - 1)^2*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)

a(n) is multiplicative with a(p) = phi(p), a(p^k) = phi(p^k)-phi(p^(k-1)) and phi(n) = A000010(n).
a(n) = Sum_{d} A007431(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 / 2 = 0.2679480769... . - Amiram Eldar, Nov 04 2022

A232930 For each complex primitive Dirichlet character chi modulo n, let f(chi) be the least positive integer k for which chi(k) is not in the set {0,1}. Then a(n) is the sum of f(chi) over all such chi.

Original entry on oeis.org

2, 3, 6, 0, 11, 8, 8, 0, 18, 5, 22, 0, 11, 12, 31, 0, 34, 17, 10, 0, 45, 20, 32, 0, 24, 17, 54, 0, 63, 24, 21, 0, 30, 20, 70, 0, 27, 22, 79, 0, 84, 27, 24, 0, 93, 20, 72, 0, 36, 33, 102, 0, 55, 38, 37, 0, 114, 27, 118, 0, 52, 48, 69, 0, 130, 47, 42, 0, 143, 40, 151, 0, 32, 55, 90, 0, 155, 52, 72, 0, 162, 33, 96, 0, 57, 56, 181, 0, 114, 63, 58, 0, 107, 40, 193, 0, 72, 48, 198, 0, 203, 78, 39, 0, 210, 60, 216, 0, 79, 60, 225, 0, 126, 85, 100, 0, 159, 46

Offset: 3

Steven Finch, Dec 02 2013

nonn

			a(6)=0 since there are no primitive Dirichlet characters mod 6.

S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
G. Martin and P. Pollack, The average least character non-residue and further variations on a theme of Erdos, J. London Math. Soc. 87 (2013) 22-42.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A007431.

A256936 Decimal expansion of Sum_{k>=1} phi(k)/2^k, where phi is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 13 2015

			1.36763080198502235079050814621308813907489199896...

Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, p. 139.

Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique, L'enseignement Mathématique, Université de Genève, 1980, p. 61.
Eric Weisstein's MathWorld, Totient Function.
Wikipedia, Euler's totient function.

Cf. A000010, A007431.

digits = 99; m0 = 10; dd = 10; Clear[f]; f[m_] := f[m] = Sum[EulerPhi[n]/2^n, {n, 1, m}] // N[#, digits + 2*dd]&; f[m = m0] ; While[RealDigits[f[2*m], 10, digits + dd ] != RealDigits[f[m], 10, digits + dd ], m = 2*m; Print[m]]; RealDigits[f[m], 10, digits] // First

suminf(n=1,eulerphi(n)/2^n) \\ Charles R Greathouse IV, Apr 20 2016

Equals Sum_{k>=1} A007431(k)/(2^k - 1). - Amiram Eldar, Jun 23 2020

cp's OEIS Frontend

A130054 Inverse Moebius transform of A023900.

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A062790 Moebius transform of the cototient function A051953.

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A354058 Square array read by ascending antidiagonals: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n.

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A354061 Irregular table read by rows: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n, 1 <= k <= psi(n), psi = A002322.

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A318839 Restricted growth sequence transform of A318838.

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A321322 a(n) = Sum_{d|n} mu(n/d)*J_2(d), where J_2() is the Jordan function (A007434).

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A340190 Möbius transform of A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

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A114810 Number of complex, weakly primitive Dirichlet characters modulo n.