A364222 -id:A364222 - cp's OEIS frontend

A373279 Expansion of Sum_{k>=0} x^(3^k) / (1 - 3*x^(3^k)).

1, 3, 10, 27, 81, 246, 729, 2187, 6571, 19683, 59049, 177174, 531441, 1594323, 4783050, 14348907, 43046721, 129140409, 387420489, 1162261467, 3486785130, 10460353203, 31381059609, 94143181014, 282429536481, 847288609443, 2541865834900, 7625597484987

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373280, A373281, A373282, A373283.

Cf. A000244, A046211, A051064, A364222.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=3) = sumdiv(n, d, d*b(d, k));

G.f. A(x) satisfies A(x) = x/(1 - 3*x) + A(x^3).
If n == 0 (mod 3), a(n) = 3^n + a(n/3) otherwise a(n) = 3^n.
a(n) = Sum_{d|n} d * A046211(d).

A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 10, 6, 7, 8, 9, 20, 11, 12, 13, 14, 30, 16, 17, 18, 19, 40, 21, 22, 23, 24, 75, 26, 27, 28, 29, 60, 31, 32, 33, 34, 70, 36, 37, 38, 39, 80, 41, 42, 43, 44, 90, 46, 47, 48, 49, 150, 51, 52, 53, 54, 110, 56, 57, 58, 59, 120, 61, 62, 63, 64, 130, 66, 67, 68, 69, 140, 71, 72, 73, 74, 225

Offset: 1

Seiichi Manyama, Jul 14 2023

Cf. A055457, A112765, A359100.

Cf. A091512, A364222, A364224.

a[n_] := n * (IntegerExponent[n, 5] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)

PARI
```
a(n) = n*(valuation(n, 5)+1);
```

a(n) = n * A055457(n).
If n == 0 (mod 5), a(n) = n + 5 * a(n/5) otherwise a(n) = n.
From Amiram Eldar, Jul 14 2023: (Start)
Multiplicative with a(5^e) = (e+1)*5^e and a(p^e) = p*e if p != 5.
Dirichlet g.f.: (5^s/(5^s-5)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (5/8)*n^2. (End)

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)

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A373279 Expansion of Sum_{k>=0} x^(3^k) / (1 - 3*x^(3^k)).

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A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.

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

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