A359212 -id:A359212 - cp's OEIS frontend

A359211 a(n) = tau(3*n-1)/2, where tau(n) = number of divisors of n, cf. A000005.

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 4, 1, 2, 2, 3, 1, 2, 2, 5, 1, 2, 1, 3, 2, 3, 1, 4, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 1, 6, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 5, 1, 4, 2, 3, 1, 2, 1, 6, 2, 2, 2, 3, 2, 2, 2, 6, 1, 4, 1, 3, 1, 3, 3, 4, 1, 2, 1, 6, 1, 4, 1

Offset: 1

Seiichi Manyama, Dec 21 2022

Also number of divisors of 3*n-1 of form 3*k+1 (or 3*k+2).

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

Cf. A078703, A099774.

Cf. A000005, A001620, A001817, A001822, A359212.

a[n_] := DivisorSigma[0, 3*n-1]/2; Array[a, 100] (* Amiram Eldar, Dec 21 2022 *)

PARI
```
a(n) = numdiv(3*n-1)/2;
```
PARI
```
a(n) = sumdiv(3*n-1, d, d%3==1);
```
PARI
```
a(n) = sumdiv(3*n-1, d, d%3==2);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(3*k-1))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(3*k-2))))

G.f.: Sum_{k>0} x^k/(1 - x^(3*k-1)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(3*k-2)).
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 2*log(3))*n/3 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 26 2022

A359239 Number of divisors of 3n-2 of form 3k+2.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 2, 1, 2, 0, 2, 0, 4, 0, 2, 0, 2, 2, 2, 0, 3, 0, 4, 0, 2, 0, 2, 2, 4, 0, 2, 0, 4, 0, 2, 0, 4, 2, 2, 1, 2, 0, 4, 0, 4, 0, 2, 2, 2, 0, 4, 0, 6, 0, 2, 0, 2, 2, 2, 0, 4, 2, 4, 0, 3, 0, 2, 2, 4, 0, 2, 0, 6, 0, 2, 0, 4, 2, 4, 0, 2, 0, 4, 2, 4, 0, 2, 2, 2

Offset: 1

Seiichi Manyama, Dec 22 2022

Cf. A001822, A359211, A359212.

Cf. A359240, A359241.

Table[Count[Divisors[3 n-2],?(IntegerQ[(#-2)/3]&)],{n,100}] (* _Harvey P. Dale, Apr 23 2023 *)
a[n_] := DivisorSum[3*n-2, 1 &, Mod[#, 3] == 2 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

PARI
```
a(n) = sumdiv(3*n-2, d, d%3==2);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(3*k-1)))))

a(n) = A001822(3*n-2).

G.f.: Sum_{k>0} x^(2*k)/(1 - x^(3*k-1)).

A363889 Sum of divisors of 3n-2 of form 3k+1.

Original entry on oeis.org

1, 5, 8, 11, 14, 21, 20, 23, 26, 40, 32, 35, 38, 55, 44, 47, 57, 70, 56, 59, 62, 85, 68, 88, 74, 100, 80, 83, 86, 115, 112, 95, 98, 140, 104, 107, 110, 168, 116, 119, 122, 160, 128, 154, 160, 175, 140, 143, 146, 190, 152, 184, 158, 231, 164, 167, 183, 220, 208, 179, 182, 235, 188

Offset: 1

Seiichi Manyama, Jun 26 2023

nonn

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

Cf. A363514, A363890, A363891.

Cf. A078181, A359212.

a[n_] := DivisorSum[3*n - 2, # &, Mod[#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
a(n) = sumdiv(3*n-2, d, (d%3==1)*d);
```

a(n) = A078181(3*n-2).

G.f.: Sum_{k>0} (3*k-2) * x^k/(1 - x^(3*k-2)).

A364064 Expansion of Sum_{k>0} k * x^k / (1 - x^(3*k-2)).

Original entry on oeis.org

1, 3, 4, 5, 6, 9, 8, 9, 10, 16, 12, 13, 14, 21, 16, 17, 21, 26, 20, 21, 22, 31, 24, 32, 26, 36, 28, 29, 30, 41, 40, 33, 34, 50, 36, 37, 38, 60, 40, 41, 42, 56, 44, 54, 56, 61, 48, 49, 50, 66, 52, 64, 54, 81, 56, 57, 63, 76, 72, 61, 62, 81, 64, 76, 66, 99, 68, 69, 70, 102, 72, 73, 88, 108, 76, 77, 78, 101

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

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

Cf. A359212, A363889.

a[n_] := DivisorSum[3*n - 2, # + 2 &, Mod[#, 3] == 1 &]/3; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)

a(n) = sumdiv(3*n-2, d, (d%3==1)*(d+2))/3;

a(n) = (1/3) * Sum_{d | 3*n-2, d==1 (mod 3)} (d+2).

G.f.: Sum_{k>0} x^k / (1 - x^(3*k-2))^2.

A363853 Number of divisors of 7n-6 of form 7k+1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Cf. A359212, A359238, A359309.

Cf. A279061.

a[n_] := DivisorSum[7*n - 6, 1 &, Mod[#, 7] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 25 2023 *)

PARI
```
a(n) = sumdiv(7*n-6, d, d%7==1);
```

a(n) = A279061(7*n-6).

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

cp's OEIS Frontend

A359211 a(n) = tau(3*n-1)/2, where tau(n) = number of divisors of n, cf. A000005.

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A359239 Number of divisors of 3*n-2 of form 3*k+2.

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A363889 Sum of divisors of 3*n-2 of form 3*k+1.

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A364064 Expansion of Sum_{k>0} k * x^k / (1 - x^(3*k-2)).

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A363853 Number of divisors of 7*n-6 of form 7*k+1.

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A359239 Number of divisors of 3n-2 of form 3k+2.

A363889 Sum of divisors of 3n-2 of form 3k+1.

A363853 Number of divisors of 7n-6 of form 7k+1.