A359241 -id:A359241 - cp's OEIS frontend

A359240 Number of divisors of 4n-3 of form 4k+3.

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

Offset: 1

Seiichi Manyama, Dec 22 2022

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

Cf. A359239, A359241.

Cf. A001842.

a[n_] := DivisorSum[4*n-3, 1 &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

PARI
```
a(n) = sumdiv(4*n-3, d, d%4==3);
```

my(N=100, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^(4*k-1)))))

a(n) = A001842(4*n-3).

G.f.: Sum_{k>0} x^(3*k)/(1 - x^(4*k-1)).

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)).

A359327 Number of divisors of 6n-5 of form 6k+5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 1, 2

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A319995, A359305, A359324, A359325, A359326.

Cf. A359239, A359240, A359241.

a[n_] := DivisorSum[6*n-5, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Aug 14 2023 *)

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

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/(1-x^(6*k-1)))))

a(n) = A319995(6*n-5).

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(6*k-1)).

A363858 Number of divisors of 7n-6 of form 7k+6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Cf. A361691, A363854, A363855, A363856, A363857.
Cf. A359239, A359241, A359327.
Cf. A363808.

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

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

a(n) = A363808(7*n-6).

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

A364103 Sum of divisors of 5n-4 of form 5k+4.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 13, 0, 0, 0, 18, 0, 0, 0, 23, 9, 0, 0, 28, 0, 0, 0, 33, 0, 23, 0, 38, 0, 0, 0, 43, 0, 0, 28, 48, 0, 0, 0, 67, 0, 0, 0, 91, 0, 0, 0, 63, 0, 0, 0, 68, 38, 33, 0, 73, 0, 0, 0, 78, 0, 43, 0, 83, 0, 0, 0, 126, 0, 0, 48, 93, 19, 0, 0, 98, 0, 0, 0, 156, 0, 43, 0, 108, 0, 0, 0, 113, 58

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364100, A364101, A364102.

Cf. A284103, A359241, A364107.

a[n_] := DivisorSum[5*n - 4, # &, Mod[#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(5*n-4, d, (d%5==4)*d);
```

a(n) = A284103(5*n-4).

G.f.: Sum_{k>0} (5*k-1) * x^(4*k) / (1 - x^(5*k-1)).

A364107 Expansion of Sum_{k>0} k * x^(4k) / (1 - x^(5k-1)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 5, 2, 0, 0, 6, 0, 0, 0, 7, 0, 5, 0, 8, 0, 0, 0, 9, 0, 0, 6, 10, 0, 0, 0, 14, 0, 0, 0, 19, 0, 0, 0, 13, 0, 0, 0, 14, 8, 7, 0, 15, 0, 0, 0, 16, 0, 9, 0, 17, 0, 0, 0, 26, 0, 0, 10, 19, 4, 0, 0, 20, 0, 0, 0, 32, 0, 9, 0, 22, 0, 0, 0, 23, 12, 0, 0, 33, 0, 0, 0

Offset: 1

Seiichi Manyama, Jul 05 2023

nonn

Cf. A364104, A364105, A364106.

Cf. A359241, A364103.

a[n_] := DivisorSum[5*n - 4, # + 1 &, Mod[#, 5] == 4 &]/5; Array[a, 100] (* Amiram Eldar, Jul 12 2023 *)

a(n) = sumdiv(5*n-4, d, (d%5==4)*(d+1))/5;

a(n) = (1/5) * Sum_{d | 5*n-4, d==4 (mod 5)} (d+1).

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

cp's OEIS Frontend

A359240 Number of divisors of 4*n-3 of form 4*k+3.

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

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A359327 Number of divisors of 6*n-5 of form 6*k+5.

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A363858 Number of divisors of 7*n-6 of form 7*k+6.

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A364103 Sum of divisors of 5*n-4 of form 5*k+4.

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A364107 Expansion of Sum_{k>0} k * x^(4*k) / (1 - x^(5*k-1)).

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A359240 Number of divisors of 4n-3 of form 4k+3.

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

A359327 Number of divisors of 6n-5 of form 6k+5.

A363858 Number of divisors of 7n-6 of form 7k+6.

A364103 Sum of divisors of 5n-4 of form 5k+4.

A364107 Expansion of Sum_{k>0} k * x^(4k) / (1 - x^(5k-1)).