A359309 -id:A359309 - cp's OEIS frontend

A359305 Number of divisors of 6n-1 of form 6k+1.

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Also number of divisors of 6*n-1 of form 6*k+5.

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

Cf. A279060, A319995.
Cf. A000005, A001620, A016969, A078703, A359211, A359233.
Cf. A359306, A359307, A359308, A359309.
Cf. A359324, A359325, A359326, A359327.

a[n_] := DivisorSigma[0, 6*n - 1]/2; Array[a, 100] (* Amiram Eldar, Dec 26 2022 *)

PARI
```
a(n) = sumdiv(6*n-1, d, d%6==1);
```
PARI
```
a(n) = sumdiv(6*n-1, d, d%6==5);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-1))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(5*k-4)/(1-x^(6*k-5))))

a(n) = A279060(6*n-1) = A319995(6*n-1).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-1)).
G.f.: Sum_{k>0} x^(5*k-4)/(1 - x^(6*k-5)).
From Amiram Eldar, Dec 26 2022: (Start)
a(n) = A000005(A016969(n-1))/2.
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 3*log(2) + 2*log(3))*n/6 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). (End)

A359306 Number of divisors of 6n-2 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359307, A359308, A359309.

a[n_] := DivisorSum[6*n-2, 1 &, Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(6*n-2, d, d%6==1);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-2))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(4*k-3)/(1-x^(6*k-5))))

a(n) = A279060(6*n-2).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-2)).
G.f.: Sum_{k>0} x^(4*k-3)/(1 - x^(6*k-5)).

A359307 Number of divisors of 6n-3 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359306, A359308, A359309.

a[n_] := DivisorSum[6*n-3, 1 &, Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(6*n-3, d, d%6==1);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-3))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(3*k-2)/(1-x^(6*k-5))))

a(n) = A279060(6*n-3).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-3)).
G.f.: Sum_{k>0} x^(3*k-2)/(1 - x^(6*k-5)).

A359308 Number of divisors of 6n-4 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359306, A359307, A359309.

a[n_] := DivisorSum[6*n-4, 1 &, Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(6*n-4, d, d%6==1);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-4))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(6*k-5))))

a(n) = A279060(6*n-4).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-4)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(6*k-5)).

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

A359305 Number of divisors of 6*n-1 of form 6*k+1.

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A359306 Number of divisors of 6*n-2 of form 6*k+1.

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A359307 Number of divisors of 6*n-3 of form 6*k+1.

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A359308 Number of divisors of 6*n-4 of form 6*k+1.

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

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A359305 Number of divisors of 6n-1 of form 6k+1.

A359306 Number of divisors of 6n-2 of form 6k+1.

A359307 Number of divisors of 6n-3 of form 6k+1.

A359308 Number of divisors of 6n-4 of form 6k+1.

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