A091512 -id:A091512 - cp's OEIS frontend

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

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)

A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

Original entry on oeis.org

1, 3, 2, 4, 3, 1, 7, 6, 4, 3, 8, 7, 5, 4, 1, 10, 9, 7, 6, 3, 2, 11, 10, 8, 7, 4, 3, 1, 15, 14, 12, 11, 8, 7, 5, 4, 16, 15, 13, 12, 9, 8, 6, 5, 1, 18, 17, 15, 14, 11, 10, 8, 7, 3, 2, 19, 18, 16, 15, 12, 11, 9, 8, 4, 3, 1, 22, 21, 19, 18, 15, 14, 12, 11, 7, 6, 4, 3

Offset: 1

Gary W. Adamson, Jul 27 2008

Row sums give A143157.
Left border gives A005187.
Right border gives A001511.

			First few rows of the triangle =
        k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=1:   1;
  n=2:   3,  2;
  n=3:   4,  3,  1;
  n=4:   7,  6,  4,  3;
  n=5:   8,  7,  5,  4,  1;
  n=6:  10,  9,  7,  6,  3,  2;
  n=7:  11, 10,  8,  7,  4,  3,  1;
  ...
Row 6 = (10, 9, 7, 6, 3, 2) = partial sums of the first 6 terms of the ruler sequence, starting from the right: (1, 2, 1, 3, 1, 2,...).

Cf. A001511, A000012, A091512, A005187.

T(n,k) = k--; 2*(n-k) - hammingweight(n) + hammingweight(k); \\ Kevin Ryde, Oct 07 2021

Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.
From Kevin Ryde, Oct 07 2021: (Start)
T(n,k) = A005187(n) - A005187(k-1).
G.f.: (V(x) - V(x*y)) * y/((1-x)*(1-y)) where V(x) is the g.f. of A001511.
(End)

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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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A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

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