A029935 -id:A029935 - cp's OEIS frontend

A143258 Triangle read by rows, A054525 * A143257, 1<=k<=n.

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 6, 2, 2, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 02 2008

Left border = phi(n), A000010.

Row sums = A029935: (1, 2, 4, 5, 8, 8, 12, 12,...)

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
4, 1, 1, 1, 1;
2, 2, 1, 1, 1, 1;
6, 1, 1, 1, 1, 1, 1;
...

Cf. A143257, A054525, A029935, A000010.

Mobius transform of triangle A143257

A328641 Dirichlet g.f.: zeta(s)^2 / zeta(s-1)^2.

Original entry on oeis.org

1, -2, -4, -1, -8, 8, -12, 0, 0, 16, -20, 4, -24, 24, 32, 1, -32, 0, -36, 8, 48, 40, -44, 0, 8, 48, 4, 12, -56, -64, -60, 2, 80, 64, 96, 0, -72, 72, 96, 0, -80, -96, -84, 20, 0, 88, -92, -4, 24, -16, 128, 24, -104, -8, 160, 0, 144, 112, -116, -32, -120, 120, 0, 3, 192

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A029935.
Dirichlet convolution of A023900 with itself.
Inverse Moebius transform of A101035.

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

Cf. A023900, A029935, A101035.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[EulerPhi[j], EulerPhi[j], j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 65}]
f[p_, e_] := If[e == 1, 2*(1 - p), (p - 1)*(e*p - p - e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv(n, d, eulerphi(d) * eulerphi(n/d))))} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA029935(n/d) * a(d).
a(n) = Sum_{d|n} A101035(d).
Multiplicative with a(p) = 2*(1-p), and a(p^e) = (p-1)*(e*p-p-e-1) for e > 1. - Amiram Eldar, Dec 03 2022

A338165 Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

Original entry on oeis.org

1, 14, 52, 161, 248, 728, 684, 1680, 2080, 3472, 2660, 8372, 4392, 9576, 12896, 16576, 9824, 29120, 13716, 39928, 35568, 37240, 24332, 87360, 46376, 61488, 74412, 110124, 48776, 180544, 59580, 157696, 138320, 137536, 169632, 334880, 101304, 192024, 228384, 416640

Offset: 1

Ilya Gutkovskiy, Oct 14 2020

Dirichlet convolution of Jordan function J_3 (A059376) with itself.

Wikipedia, Jordan's totient function

Cf. A000005, A007427, A029935, A059376, A338164.

Jordan3[n_] := Sum[d^3 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan3[d] Jordan3[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]
a[1] = 1; f[p_, e_] := p^(3 e - 6) (p^6 + e (p^3 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 40}]

Multiplicative with a(p^e) = p^(3*e - 6) * (p^6 + e * (p^3 - 1)^2 - 1).
a(n) = Sum_{d|n} J_3(d) * J_3(n/d).
a(n) = Sum_{d|n} d^3 * tau(d) * A007427(n/d), where tau = A000005.
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^3.
Sum_{k=1..n} a(k) ~ 2025 * n^4 * ((log(n) + 2*gamma - 1/4)/Pi^8 - 180*zeta'(4) / Pi^12), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020

A349692 Dirichlet convolution of the gcd-sum function (A018804) with itself.

Original entry on oeis.org

1, 6, 10, 25, 18, 60, 26, 88, 67, 108, 42, 250, 50, 156, 180, 280, 66, 402, 74, 450, 260, 252, 90, 880, 211, 300, 372, 650, 114, 1080, 122, 832, 420, 396, 468, 1675, 146, 444, 500, 1584, 162, 1560, 170, 1050, 1206, 540, 186, 2800, 435, 1266, 660, 1250, 210, 2232, 756

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Vaclav Kotesovec, Asymptotics of Sum_{k=1..n} a(k) with a graph

Cf. A000203, A018804, A029935, A038040, A344683.

A018804[n_] := Sum[GCD[n,k], {k, 1, n}]; a[n_] := Sum[A018804[d] A018804[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e + 1)*p^(e - 2)*((e + 2)*(e + 3)*p^2 - 2*e*(e + 2)*p + e*(e - 1))/6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 25 2021 *)

A029935(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ From A029935.
A349692(n) = sumdiv(n, d, A029935(n/d)*d*numdiv(d)); \\ Antti Karttunen, Nov 25 2021

Dirichlet g.f.: zeta(s-1)^4 / zeta(s)^2.
a(n) = Sum_{d|n} A018804(d) * A018804(n/d).
a(n) = Sum_{d|n} A029935(d) * A038040(n/d).
a(n) = Sum_{d|n} A344683(d) * A000203(n/d).
Multiplicative with a(p^e) = (e+1) * p^(e-2) * ((e+2)*(e+3)*p^2 - 2*e*(e+2)*p + e*(e-1))/6. - Amiram Eldar, Nov 25 2021

A349741 a(n) = Product_{k=1..n-1} phi(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 1, 32, 1, 6, 256, 16, 1, 96, 1, 1024, 2304, 10, 1, 16384, 256, 12, 2304, 13824, 1, 524288, 1, 2048, 102400, 16, 5308416, 14155776, 1, 18, 589824, 134217728, 1, 63700992, 1, 1024000, 86973087744, 22, 1, 8589934592, 46656, 1310720

Offset: 1

Ilya Gutkovskiy, Nov 28 2021

nonn

Cf. A000010, A001088, A029935, A029940, A046022 (positions of 1's), A051190.

Table[Product[EulerPhi[GCD[n, k]], {k, 1, n - 1}], {n, 1, 50}]

a(n) = prod(k=1, n-1, eulerphi(gcd(n, k))); \\ Michel Marcus, Nov 28 2021

a(n) = Product_{d|n, d < n} phi(d)^phi(n/d).

A331376 Dirichlet convolution of the reduced totient function with itself.

Original entry on oeis.org

1, 2, 4, 5, 8, 8, 12, 8, 16, 16, 20, 16, 24, 24, 24, 16, 32, 32, 36, 32, 36, 40, 44, 24, 56, 48, 60, 48, 56, 48, 60, 32, 60, 64, 72, 60, 72, 72, 72, 48, 80, 72, 84, 80, 88, 88, 92, 44, 120, 112, 96, 96, 104, 120, 120

Offset: 1

Torlach Rush, Jan 14 2020

nonn

Differs from A029935 when A002322 differs from A000010.

Cf. A000010, A029935, A002322.

a(n) = sumdiv(n, d, lcm(znstar(d)[2]) * lcm(znstar(n/d)[2]));

a(n) = Sum{d|n} A002322(d) * A002322(n/d).

cp's OEIS Frontend

A143258 Triangle read by rows, A054525 * A143257, 1<=k<=n.

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A328641 Dirichlet g.f.: zeta(s)^2 / zeta(s-1)^2.

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A338165 Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

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A349692 Dirichlet convolution of the gcd-sum function (A018804) with itself.

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A349741 a(n) = Product_{k=1..n-1} phi(gcd(n,k)).

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A331376 Dirichlet convolution of the reduced totient function with itself.

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