A363807 -id:A363807 - cp's OEIS frontend

A363795 Number of divisors of n of the form 7*k + 2.

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

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363805, A363806, A363807, A363808.
Cf. A001822, A001877.
Cf. A284443.
Cf. A001620, A016630, A354628 (psi(2/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==2);
```

G.f.: Sum_{k>0} x^(2*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-5)/(1 - x^(7*k-5)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,7) - (1 - gamma)/7 = 0.188117..., gamma(2,7) = -(psi(2/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363805 Number of divisors of n of the form 7*k + 3.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363806, A363807, A363808.
Cf. A001842, A001878.
Cf. A284444.
Cf. A001620, A016630, A354629 (psi(3/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==3);
```

G.f.: Sum_{k>0} x^(3*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-4)/(1 - x^(7*k-4)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,7) - (1 - gamma)/7 = -0.0004108181..., gamma(3,7) = -(psi(3/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363806 Number of divisors of n of the form 7*k + 4.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363805, A363807, A363808.
Cf. A284445.
Cf. A001620, A016630, A354630 (psi(4/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 4 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==4);
```

G.f.: Sum_{k>0} x^(4*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-3)/(1 - x^(7*k-3)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,7) - (1 - gamma)/7 = -0.102846..., gamma(4,7) = -(psi(4/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363808 Number of divisors of n of the form 7*k + 6.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363805, A363806, A363807.
Cf. A284105.
Cf. A001620, A016630, A354632 (psi(6/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 6 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

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

G.f.: Sum_{k>0} x^(6*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-1)/(1 - x^(7*k-1)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(6,7) - (1 - gamma)/7 = -0.218328..., gamma(6,7) = -(psi(6/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363857 Number of divisors of 7n-5 of form 7k+6.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Also number of divisors of 7*n-5 of form 7*k+5.

Cf. A361691, A363854, A363855, A363856, A363858.

Cf. A363807, A363808.

a[n_] := DivisorSum[7*n - 5, 1 &, Mod[#, 7] == 6 &]; Array[a, 100] (* Amiram Eldar, Jun 25 2023 *)

PARI
```
a(n) = sumdiv(7*n-5, d, d%7==6);
```

a(n) = A363807(7*n-5) = A363808(7*n-5).
G.f.: Sum_{k>0} x^(5*k)/(1 - x^(7*k-1)).
G.f.: Sum_{k>0} x^(6*k-1)/(1 - x^(7*k-2)).

A363886 Number of divisors of 7n-3 of form 7k+5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 26 2023

nonn

Cf. A363850, A363855, A363885.

Cf. A363807.

a[n_] := DivisorSum[7*n - 3, 1 &, Mod[#, 7] == 5 &]; Array[a, 100] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
a(n) = sumdiv(7*n-3, d, d%7==5);
```

a(n) = A363807(7*n-3).

G.f.: Sum_{k>0} x^(5*k-1)/(1 - x^(7*k-2)).

A362845 Number of divisors of 7n-2 of form 7k+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Also number of divisors of 7*n-2 of form 7*k+5.

Cf. A279061, A363807.

a[n_] := DivisorSum[7*n - 2, 1 &, Mod[#, 7] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 25 2023 *)

PARI
```
a(n) = sumdiv(7*n-2, d, d%7==1);
```

a(n) = A279061(7*n-2) = A363807(7*n-2).
G.f.: Sum_{k>0} x^(5*k-4)/(1 - x^(7*k-6)).
G.f.: Sum_{k>0} x^k/(1 - x^(7*k-2)).

A363860 Number of divisors of 7n-1 of form 7k+4.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Also number of divisors of 7*n-1 of form 7*k+5.

Cf. A363806, A363807.

a[n_] := DivisorSum[7*n - 1, 1 &, Mod[#, 7] == 4 &]; Array[a, 100] (* Amiram Eldar, Jun 25 2023 *)

PARI
```
a(n) = sumdiv(7*n-1, d, d%7==4);
```

a(n) = A363806(7*n-1) = A363807(7*n-1).
G.f.: Sum_{k>0} x^(5*k-2)/(1 - x^(7*k-3)).
G.f.: Sum_{k>0} x^(4*k-1)/(1 - x^(7*k-2)).

A363878 Number of divisors of 7n-4 of form 7k+2.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 25 2023

nonn

Also number of divisors of 7*n-4 of form 7*k+5.

Cf. A363795, A363807.

a[n_] := DivisorSum[7*n - 4, 1 &, Mod[#, 7] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 25 2023 *)

PARI
```
a(n) = sumdiv(7*n-4, d, d%7==2);
```

a(n) = A363795(7*n-4) = A363807(7*n-4).
G.f.: Sum_{k>0} x^(5*k-3)/(1 - x^(7*k-5)).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(7*k-2)).

A363880 Number of divisors of 7n-6 of form 7k+3.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 25 2023

nonn

Also number of divisors of 7*n-6 of form 7*k+5.

Cf. A363805, A363807.

a[n_] := DivisorSum[7*n - 6, 1 &, Mod[#, 7] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 25 2023 *)

PARI
```
a(n) = sumdiv(7*n-6, d, d%7==3);
```

a(n) = A363805(7*n-6) = A363807(7*n-6).
G.f.: Sum_{k>0} x^(5*k-2)/(1 - x^(7*k-4)).
G.f.: Sum_{k>0} x^(3*k)/(1 - x^(7*k-2)).

cp's OEIS Frontend

A363795 Number of divisors of n of the form 7*k + 2.

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A363805 Number of divisors of n of the form 7*k + 3.

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A363806 Number of divisors of n of the form 7*k + 4.

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A363808 Number of divisors of n of the form 7*k + 6.

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A363857 Number of divisors of 7*n-5 of form 7*k+6.

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A363886 Number of divisors of 7*n-3 of form 7*k+5.

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A362845 Number of divisors of 7*n-2 of form 7*k+1.

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A363860 Number of divisors of 7*n-1 of form 7*k+4.

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A363857 Number of divisors of 7n-5 of form 7k+6.

A363886 Number of divisors of 7n-3 of form 7k+5.

A362845 Number of divisors of 7n-2 of form 7k+1.

A363860 Number of divisors of 7n-1 of form 7k+4.

A363878 Number of divisors of 7n-4 of form 7k+2.

A363880 Number of divisors of 7n-6 of form 7k+3.