A359099 -id:A359099 - cp's OEIS frontend

A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

1, 2, 3, 4, 6, 6, 7, 8, 9, 12, 11, 12, 13, 14, 18, 16, 17, 18, 19, 24, 21, 22, 23, 24, 31, 26, 27, 28, 29, 36, 31, 32, 33, 34, 42, 36, 37, 38, 39, 48, 41, 42, 43, 44, 54, 46, 47, 48, 49, 62, 51, 52, 53, 54, 66, 56, 57, 58, 59, 72, 61, 62, 63, 64, 78, 66, 67, 68, 69, 84, 71, 72, 73, 74, 93, 76

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, Totient Function.

Cf. A129527, A327625, A359099, A373188.

Cf. A000010, A055457.

Array[DivisorSum[#, EulerPhi[5 #] &]/4 &, 76] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 5, (5^(e + 1) - 1)/4, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

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

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(5*k)*x^k/(1-x^k))/4)

G.f.: Sum_{k>=1} phi(5 * k) * x^k / (4 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(5^k) / (1 - x^(5^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = p if p != 5.
Dirichlet g.f.: zeta(s-1)*(1+1/(5^s-1)).
Sum_{k=1..n} a(k) ~ (25/48) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^5).
If n == 0 (mod 5), a(n) = n + a(n/5) otherwise a(n) = n. (End)

A359102 a(n) = phi(7 * n)/6.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 4, 6, 4, 10, 4, 12, 7, 8, 8, 16, 6, 18, 8, 14, 10, 22, 8, 20, 12, 18, 14, 28, 8, 30, 16, 20, 16, 28, 12, 36, 18, 24, 16, 40, 14, 42, 20, 24, 22, 46, 16, 49, 20, 32, 24, 52, 18, 40, 28, 36, 28, 58, 16, 60, 30, 42, 32, 48, 20, 66, 32, 44, 28, 70, 24, 72, 36, 40, 36, 70, 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, A359101.

Cf. A008653, A359099.

Array[EulerPhi[7 #]/6 &, 79] (* Michael De Vlieger, Dec 16 2022 *)

PARI
```
a(n) = eulerphi(7*n)/6;
```

my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(7*k)*x^k/(1-x^k)^2))

G.f.: -Sum_{k>=1} mu(7 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(7^e) = 7^e, and a(p^e) = (p-1)*p^(e-1) if p != 7.
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/7^s)).
Sum_{k=1..n} a(k) ~ (49/(16*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).

A364224 Expansion of Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 14, 8, 9, 10, 11, 12, 13, 28, 15, 16, 17, 18, 19, 20, 42, 22, 23, 24, 25, 26, 27, 56, 29, 30, 31, 32, 33, 34, 70, 36, 37, 38, 39, 40, 41, 84, 43, 44, 45, 46, 47, 48, 147, 50, 51, 52, 53, 54, 55, 112, 57, 58, 59, 60, 61, 62, 126, 64, 65, 66, 67, 68, 69, 140, 71, 72, 73, 74, 75, 76, 154

Offset: 1

Seiichi Manyama, Jul 14 2023

Cf. A214411, A359099.

Cf. A091512, A364222, A364223.

a[n_] := n * (IntegerExponent[n, 7] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)

PARI
```
a(n) = n*(valuation(n, 7)+1);
```

a(n) = n * (A214411(n) + 1).
If n == 0 (mod 7), a(n) = n + 7 * a(n/7) otherwise a(n) = n.
From Amiram Eldar, Jul 14 2023: (Start)
Multiplicative with a(7^e) = (e+1)*7^e and a(p^e) = p*e if p != 7.
Dirichlet g.f.: (7^s/(7^s-7)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (7/12)*n^2. (End)

A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 11, 14, 13, 14, 15, 16, 17, 21, 19, 20, 21, 22, 23, 28, 25, 26, 27, 28, 29, 35, 31, 32, 33, 34, 35, 43, 37, 38, 39, 40, 41, 49, 43, 44, 45, 46, 47, 56, 49, 50, 51, 52, 53, 63, 55, 56, 57, 58, 59, 70, 61, 62, 63, 64, 65, 77, 67, 68, 69, 70, 71, 86, 73, 74, 75, 76, 77, 91

Offset: 1

Seiichi Manyama, Jun 04 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A129527, A327625, A359099, A359100, A373188.

Cf. A373216.

G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^6).

If n == 0 (mod 6), a(n) = n + a(n/6) otherwise a(n) = n.

cp's OEIS Frontend

A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

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Formula

A359102 a(n) = phi(7 * n)/6.

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PARI

Formula

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

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PARI

Formula

A364224 Expansion of Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k))^2.

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Mathematica

PARI

Formula

A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

Views

Author

Keywords

Links

Crossrefs

Formula

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