A359102 -id:A359102 - cp's OEIS frontend

A372673 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = phi(k*n) / phi(k).

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 3, 4, 4, 1, 1, 2, 2, 4, 2, 1, 2, 2, 4, 4, 4, 6, 1, 1, 3, 2, 4, 3, 6, 4, 1, 2, 2, 4, 5, 4, 6, 8, 6, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 1, 2, 2, 4, 4, 4, 7, 8, 6, 8, 10, 8, 12, 1, 1, 3, 2, 5, 3, 6, 4, 9, 5, 10, 6, 12, 6

Offset: 1

Seiichi Manyama, May 10 2024

			Square array T(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...
  2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, ...
  2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, ...
  4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, ...
  2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 6, ...
  6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, ...
  4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, ...
  6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..18 give: A000010, A062570, A195459, A062570, A359101, A372671, A359102, A062570, A195459, A372672, A372676, A372671, A372677, A372678, A372679, A062570, A372681, A372671.
Main diagonal gives A000027.
Cf. A005117, A372619.

PARI
```
T(n, k) = eulerphi(k*n)/eulerphi(k);
```

A359101 a(n) = phi(5 * n)/4.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 4, 6, 5, 10, 4, 12, 6, 10, 8, 16, 6, 18, 10, 12, 10, 22, 8, 25, 12, 18, 12, 28, 10, 30, 16, 20, 16, 30, 12, 36, 18, 24, 20, 40, 12, 42, 20, 30, 22, 46, 16, 42, 25, 32, 24, 52, 18, 50, 24, 36, 28, 58, 20, 60, 30, 36, 32, 60, 20, 66, 32, 44, 30, 70, 24, 72, 36, 50, 36, 60, 24, 78

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A195459, A359102.

Cf. A008653, A359100.

Array[EulerPhi[5 #]/4 &, 79] (* Michael De Vlieger, Dec 16 2022 *)

PARI
```
a(n) = eulerphi(5*n)/4;
```

my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(5*k)*x^k/(1-x^k)^2))

G.f.: -Sum_{k>=1} mu(5 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = 5^e, and a(p^e) = (p-1)*p^(e-1) if p != 5.
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/5^s)).
Sum_{k=1..n} a(k) ~ (25/(8*Pi^2)) * n^2. (End)

A364212 a(n) = (1/(6n)) Sum_{d|n} 7^(n/d-1) * phi(7*d).

Original entry on oeis.org

1, 4, 17, 88, 481, 2812, 16808, 102988, 640545, 4035604, 25679569, 164778696, 1064714401, 6920652008, 45214871857, 296722645888, 1954878268801, 12923917765876, 85705978837393, 569944761286648, 3799631728468936, 25388448380261788, 169992219503608177, 1140364472585830196

Offset: 1

Seiichi Manyama, Jul 13 2023

nonn

Cf. A000010, A359099, A359102.

Cf. A000013, A364210, A364211.

a[n_] := DivisorSum[n, 7^(n/#-1)*EulerPhi[7*#]/(6*n) &]; Array[a, 25] (* Amiram Eldar, Jul 14 2023 *)

a(n) = sumdiv(n, d, 7^(n/d-1)*eulerphi(7*d))/(6*n);

G.f.: (-1/6) * Sum_{k>0} phi(7*k) * log(1-7*x^k)/(7*k).

A372638 a(n) = (1/6) * Sum_{k=1..n} phi(7*k).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 19, 23, 29, 33, 43, 47, 59, 66, 74, 82, 98, 104, 122, 130, 144, 154, 176, 184, 204, 216, 234, 248, 276, 284, 314, 330, 350, 366, 394, 406, 442, 460, 484, 500, 540, 554, 596, 616, 640, 662, 708, 724, 773, 793, 825, 849, 901, 919, 959, 987, 1023

Offset: 1

Seiichi Manyama, May 08 2024

nonn

Eric Weisstein's World of Mathematics, Totient Function.

Column k=7 of A372619.
Partial sums of A359102.
Cf. A000010.

Accumulate[Table[EulerPhi[7*n], {n, 1, 60}]]/6 (* Amiram Eldar, May 08 2024 *)

PARI
```
a(n) = sum(k=1, n, eulerphi(7*k))/6;
```

a(n) ~ (49/(16*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

cp's OEIS Frontend

A372673 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = phi(k*n) / phi(k).

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PARI

A359101 a(n) = phi(5 * n)/4.

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Author

Keywords

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Programs

Mathematica

PARI

PARI

Formula

A364212 a(n) = (1/(6n)) Sum_{d|n} 7^(n/d-1) * phi(7*d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A372638 a(n) = (1/6) * Sum_{k=1..n} phi(7*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A372673 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = phi(k*n) / phi(k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A359101 a(n) = phi(5 * n)/4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A364212 a(n) = (1/(6*n)) * Sum_{d|n} 7^(n/d-1) * phi(7*d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A372638 a(n) = (1/6) * Sum_{k=1..n} phi(7*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364212 a(n) = (1/(6n)) Sum_{d|n} 7^(n/d-1) * phi(7*d).