A007431 -id:A007431 - cp's OEIS frontend

A318837 Restricted growth sequence transform of A318836.

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

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

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

Cf. A007431, A062790, A318836.

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));
A318836(n) = { my(m=1); fordiv(n,d,if((dA007431(d)!=0),m *= prime(A007431(d)))); (m); }; \\
v318837 = rgs_transform(vector(up_to,n,A318836(n)));
A318837(n) = v318837[n];

For all i, j: a(i) = a(j) => A062790(i) = A062790(j).

A332685 a(n) = Sum_{k=1..n} mu(k/gcd(n, k)).

Original entry on oeis.org

1, 2, 1, 2, 0, 2, 0, 0, -1, 0, 0, 0, -1, -2, -2, -3, 0, -4, -1, -5, -4, -2, 0, -8, -3, -4, -4, -7, 0, -8, -2, -10, -5, -4, -4, -13, 0, -5, -4, -13, 1, -15, -1, -9, -10, -5, -1, -22, -4, -12, -5, -11, -1, -19, -6, -17, -6, -4, 1, -28, 0, -8, -12, -18, -6, -19, 0, -12, -5, -17

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

sign

Inverse Moebius transform of A112399.

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A002321, A007431, A007947, A008683, A063659, A112399, A276833, A304575.

Table[Sum[MoebiusMu[k/GCD[n, k]], {k, 1, n}], {n, 1, 70}]

a(n) = sum(k=1, n, moebius(k/gcd(n, k))); \\ Michel Marcus, Feb 21 2020

a(n) = Sum_{k=1..n} mu(lcm(n, k)/n).

a(n) = Sum_{d|n} A112399(d).

A007432 Moebius transform applied thrice to natural numbers.

Original entry on oeis.org

1, -1, 0, 1, 2, 0, 4, 1, 3, -2, 8, 0, 10, -4, 0, 2, 14, -3, 16, 2, 0, -8, 20, 0, 13, -10, 8, 4, 26, 0, 28, 4, 0, -14, 8, 3, 34, -16, 0, 2, 38, 0, 40, 8, 6, -20, 44, 0, 31, -13, 0, 10, 50, -8, 16, 4, 0, -26, 56, 0, 58, -28, 12, 8, 20, 0, 64, 14

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (108*n^2/Pi^6) for n = 1..1000000
N. J. A. Sloane, Transforms

Cf. A007431.

a[p_, e_] := Sum[ (-1)^k*Binomial[3, k]*p^(e - k), {k, 0, Min[e, 3]}]; a[n_] := Times @@ Apply[a, FactorInteger[n], {1}]; a[1] = 1; Table[ a[n], {n, 1, 68}] (* Jean-François Alcover, Dec 28 2011, after formula *)

Multiplicative with a(p^e) = Sum_{k=0..3} (-1)^k C(3,k)*p^(e-k)[e>=k];
Dirichlet g.f.: zeta(s-1)/zeta^3(s).
a(n) = Sum{d|n} tau_{-3}(d)*n/d = Sum{d|n} tau_{-2}(d)*phi(n/d), where tau_{-3} is A007428 and tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
Sum_{k=1..n} a(k) ~ 108 * n^2 / Pi^6. - Vaclav Kotesovec, Nov 04 2018

A341635 a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

Original entry on oeis.org

1, -2, -3, 1, -5, 6, -7, 0, 2, 10, -11, -3, -13, 14, 15, 0, -17, -4, -19, -5, 21, 22, -23, 0, 4, 26, 0, -7, -29, -30, -31, 0, 33, 34, 35, 2, -37, 38, 39, 0, -41, -42, -43, -11, -10, 46, -47, 0, 6, -8, 51, -13, -53, 0, 55, 0, 57, 58, -59, 15, -61, 62, -14, 0, 65

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Dirichlet inverse of A003967.
Moebius transform of A097945.
From Vaclav Kotesovec, Feb 19 2021: (Start)
Abs(a(n)) <= n.
a(n) = n iff n is in A030229. (End)

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

Cf. A000010, A003967, A007427, A007431, A008683, A030229 (fixed points), A046099 (positions of 0's), A068341, A097945, A276833.

Table[Sum[EulerPhi[d] MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 65}]
Table[Sum[MoebiusMu[GCD[n, k]] MoebiusMu[n/GCD[n, k]], {k, n}], {n, 65}]

a(n) = sumdiv(n, d, eulerphi(d)*moebius(d)*moebius(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} mu(gcd(n,k)) * mu(n/gcd(n,k)).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A003967(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A097945(d).
Multiplicative with a(p^e) = -p if e=1, p-1 if e=2, and 0 otherwise. - Amiram Eldar, Feb 19 2021

A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

Original entry on oeis.org

1, 4, 10, 17, 28, 40, 54, 70, 94, 112, 130, 170, 180, 216, 280, 284, 304, 376, 378, 476, 540, 520, 550, 700, 716, 720, 858, 918, 868, 1120, 990, 1144, 1300, 1216, 1512, 1598, 1404, 1512, 1800, 1960, 1720, 2160, 1890, 2210, 2632, 2200, 2254, 2840, 2682, 2864, 3040, 3060, 2860, 3432, 3640

Offset: 1

Ilya Gutkovskiy, Feb 19 2021

Dirichlet convolution of Euler totient function phi (A000010) with Jordan function J_2 (A007434).

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

Cf. A000010, A000203, A001615, A002618, A007427, A007431, A007434, A008683, A023900, A029935, A064987, A069097, A321322, A322577, A338164.

Jordan2[n_] := Sum[MoebiusMu[n/d] d^2, {d, Divisors[n]}]; a[n_] := Sum[EulerPhi[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 55}]
f[p_, e_] := p^(e-3)*(p-1)*(p^e*(p+1)^2-p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)

J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434
a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ Michel Marcus, Feb 20 2021

Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.
a(n) = Sum_{k=1..n} J_2(gcd(n,k)).
a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).
a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).
a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).
a(n) = Sum_{d|n} d * A321322(n/d).
a(n) = Sum_{d|n} d * A023900(d) * A338164(n/d).
a(n) = Sum_{d|n} d^2 * A007431(n/d).
a(n) = Sum_{d|n} mu(n/d) * A069097(d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - Vaclav Kotesovec, Feb 20 2021
a(n) = Sum_{k=1..n} J_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = Sum_{1 <= i, j <= n}  phi(gcd(i, j, n)). - Peter Bala, Jan 21 2024
Multiplicative with a(p^e) = p^(e-3)*(p-1)*(p^e*(p+1)^2-p). - Amiram Eldar, May 31 2024

A361707 Moebius transform applied twice to primes.

Original entry on oeis.org

2, -1, 1, 3, 7, 5, 13, 8, 15, 9, 27, 10, 37, 11, 23, 22, 55, 8, 63, 18, 37, 19, 79, 12, 77, 21, 62, 32, 105, -5, 123, 44, 73, 23, 101, 23, 153, 31, 83, 44, 175, 7, 187, 60, 84, 35, 207, 38, 195, 20, 113, 72, 237, 18, 181, 76, 133, 55, 273, 34, 279, 41, 148, 102, 217

Offset: 1

Ilya Gutkovskiy, Mar 21 2023

Winston de Greef, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000040, A007427, A007431, A007444, A361706.

a:= (proc(p) proc(n) uses numtheory;
       add(p(d)*mobius(n/d), d=divisors(n))
     end end@@2)(ithprime):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2023

A007427[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d], {d, Divisors[n]}]; a[n_] := Sum[A007427[n/d] Prime[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]

A007427(n) = if( n<1, 0, direuler(p=2, n, (1 - 'x)^2)[n]) \\ from Michael Somos, Nov 15 2002
A361707(n)=sumdiv(n, d, A007427(n/d) * prime(d)) \\ Winston de Greef, Mar 23 2023

a(n) = Sum_{d|n} A007427(n/d) * prime(d).

A127173 T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.

Original entry on oeis.org

1, -2, 1, -2, 0, 1, 1, -2, 0, 1, -2, 0, 0, 0, 1, 4, -2, -2, 0, 0, 1, -2, 0, 0, 0, 0, 0, 1, 0, 1, 0, -2, 0, 0, 0, 1, 1, 0, -2, 0, 0, 0, 0, 0, 1, 4, -2, 0, 0, -2, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 4, 1, -2, 0, -2, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

			First few rows of the triangle:
   1;
  -2,  1;
  -2,  0,  1;
   1, -2,  0,  1;
  -2,  0,  0,  0,  1;
   4, -2, -2,  0,  0, 1;
  -2,  0,  0,  0,  0, 0, 1;
   0,  1,  0, -2,  0, 0, 0, 1;
   1,  0, -2,  0,  0, 0, 0, 0, 1;
   4, -2,  0,  0, -2, 0, 0, 0, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A008683.
Column 1 is A007427.
Cf. A054525, A007431, A007428.

\\ here b(n) is A007427(n).
b(n)={sumdiv(n, d, moebius(d) * moebius(n/d))}
T(n,k)={if(n%k==0, b(n/k), 0)} \\ Andrew Howroyd, Feb 20 2022

Square of A054525 as lower triangular matrix.
A007431(n) = Sum_{k=1, n} k*T(n,k).
A007428(n) = Sum_{k=1..n} mu(k)*T(n,k).

Missing a(10)-a(14) and a(56) and beyond from Andrew Howroyd, Feb 20 2022

A143276 Triangle read by rows: A054525 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...).

Left border = A007431: (1, 0, 1, 1, 3, 0, 5, 2, 4,...).

			First few rows of the triangle =
1;
0, 1;
1, 0, 1;
1, 0, 0, 1;
3, 0, 0, 0, 1;
0, 1, 0, 0, 0, 1;
5, 0, 0, 0, 0, 0, 1;
2, 1, 0, 0, 0, 0, 0, 1;
...

Cf. A000010, A007431, A054523, A054525.

Moebius transform of triangle A054523.

a(17), a(18) corrected by Georg Fischer, May 29 2023

A326828 a(n) = (1/2) * Sum_{d|n} mu(n/d) * phi(d) * (psi(d) + 1), where mu = A008683, phi = A000010 and psi = A001615.

Original entry on oeis.org

1, 1, 4, 5, 13, 7, 26, 19, 34, 23, 64, 32, 89, 47, 82, 74, 151, 64, 188, 105, 167, 119, 274, 127, 296, 167, 294, 214, 433, 161, 494, 292, 421, 287, 548, 290, 701, 359, 590, 417, 859, 329, 944, 540, 742, 527, 1126, 506, 1170, 576, 1012, 757, 1429, 576, 1382

Offset: 1

Ilya Gutkovskiy, Oct 20 2019

nonn

Moebius transform applied twice to triangular numbers (A000217).

Cf. A000010, A000217, A001615, A007427, A007431, A007438, A008683, A321322, A326826.

with(numtheory):
b:= proc(n) option remember;
       add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
    end:
a:= proc(n) option remember;
       add(mobius(n/d)*b(d), d=divisors(n))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 20 2019

Table[1/2 Sum[MoebiusMu[n/d] EulerPhi[d] (DirichletConvolve[j, MoebiusMu[j]^2, j, d] + 1), {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSum[n/d, MoebiusMu[#] MoebiusMu[(n/d)/#] &], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[MoebiusMu[j] MoebiusMu[i] x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{i>=1} Sum_{j>=1} mu(j) * mu(i) * x^(i*j) / (1 - x^(i*j))^3.
Dirichlet g.f.: (zeta(s-1) + zeta(s-2)) / (2 * zeta(s)^2).
a(n) = (1/2) * Sum_{d|n} mu(n/d) * (phi(d) + J_2(d)), where J_2 = A007434.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * A007427(n/d).
a(n) = Sum_{d|n} mu(n/d) * A007438(d).
Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)^2). - Vaclav Kotesovec, Dec 11 2021

A331388 a(n) = Sum_{k=1..n} mu(gcd(n, k)) * k / gcd(n, k).

Original entry on oeis.org

1, 0, 2, 3, 9, 3, 20, 12, 24, 10, 54, 15, 77, 21, 48, 48, 135, 24, 170, 57, 103, 55, 252, 60, 240, 78, 216, 123, 405, 47, 464, 192, 273, 136, 390, 144, 665, 171, 388, 228, 819, 102, 902, 327, 456, 253, 1080, 240, 1008, 240, 678, 465, 1377, 216, 1036, 492, 853, 406, 1710

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Moebius transform of A023896.

Cf. A007427, A007431, A008683, A023896, A057661, A063445.

[&+[MoebiusMu(Gcd(n,k))*(k div Gcd(n,k)):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[MoebiusMu[GCD[n, k]] k/GCD[n, k], {k, 1, n}], {n, 1, 65}]
A023896[n_] := Sum[If[GCD[n, k] == 1, k, 0], {k, 1, n}]; Table[Sum[MoebiusMu[n/d] A023896[d], {d, Divisors[n]}], {n, 1, 65}]

a(n) = (1/n) * Sum_{k=1..n} mu(gcd(n, k)) * lcm(n, k).
a(n) = Sum_{d|n} mu(n/d) * A023896(d).
a(n) = Sum_{d|n} A007427(n/d) * A057661(d).
Sum_{k=1..n} a(k) ~ n^3 / (Pi^2 * Zeta(3)). - Vaclav Kotesovec, Feb 19 2020

cp's OEIS Frontend

A318837 Restricted growth sequence transform of A318836.

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A332685 a(n) = Sum_{k=1..n} mu(k/gcd(n, k)).

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A007432 Moebius transform applied thrice to natural numbers.

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A341635 a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

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A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

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A361707 Moebius transform applied twice to primes.

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A127173 T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.

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A143276 Triangle read by rows: A054525 * A054523 as infinite lower triangular matrices.

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