A279061 -id:A279061 - 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

A363807 Number of divisors of n of the form 7*k + 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 2, 0, 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, A363795, A363805, A363806, A363808.
Cf. A284446, A319995.
Cf. A001620, A016630, A354631 (psi(5/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 5 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

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

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-2)/(1 - x^(7*k-2)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,7) - (1 - gamma)/7 = -0.169787..., gamma(5,7) = -(psi(5/7) + 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

A361691 Number of divisors of 7n-1 of form 7k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Also number of divisors of 7*n-1 of form 7*k+6.

Cf. A363854, A363855, A363856, A363857, A363858.

Cf. A279061, A363808.

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

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

a(n) = A279061(7*n-1) = A363808(7*n-1).
G.f.: Sum_{k>0} x^(6*k-5)/(1 - x^(7*k-6)).
G.f.: Sum_{k>0} x^k/(1 - x^(7*k-1)).

A363850 Number of divisors of 7n-3 of form 7k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

Also number of divisors of 7*n-3 of form 7*k+4.

Cf. A279061, A363806.

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

PARI
```
a(n) = sumdiv(7*n-3, d, d%7==1);
```

a(n) = A279061(7*n-3) = A363806(7*n-3).
G.f.: Sum_{k>0} x^(4*k-3)/(1 - x^(7*k-6)).
G.f.: Sum_{k>0} x^k/(1 - x^(7*k-3)).

A363852 Number of divisors of 7n-5 of form 7k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 24 2023

nonn

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

Cf. A279061, A363795.

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

PARI
```
a(n) = sumdiv(7*n-5, d, d%7==1);
```

a(n) = A279061(7*n-5) = A363795(7*n-5).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(7*k-6)).
G.f.: Sum_{k>0} x^k/(1 - x^(7*k-5)).

A373336 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4k) + x^(5k) + x^(6*k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 01 2024

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

Cf. A002324, A048272, A373335.

Cf. A279061, A363795.

my(N=110, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1-x^(7*k))))

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

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

a(n) = A279061(n) - A363795(n).

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)).

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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A363807 Number of divisors of n of the form 7*k + 5.

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

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A361691 Number of divisors of 7*n-1 of form 7*k+1.

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A363850 Number of divisors of 7*n-3 of form 7*k+1.

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A363852 Number of divisors of 7*n-5 of form 7*k+1.

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A361691 Number of divisors of 7n-1 of form 7k+1.

A363850 Number of divisors of 7n-3 of form 7k+1.

A363852 Number of divisors of 7n-5 of form 7k+1.

A373336 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4k) + x^(5k) + x^(6*k)).

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