cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A372713 Number of divisors of 3n; a(n) = tau(3*n) = A000005(3*n).

Original entry on oeis.org

2, 4, 3, 6, 4, 6, 4, 8, 4, 8, 4, 9, 4, 8, 6, 10, 4, 8, 4, 12, 6, 8, 4, 12, 6, 8, 5, 12, 4, 12, 4, 12, 6, 8, 8, 12, 4, 8, 6, 16, 4, 12, 4, 12, 8, 8, 4, 15, 6, 12, 6, 12, 4, 10, 8, 16, 6, 8, 4, 18, 4, 8, 8, 14, 8, 12, 4, 12, 6, 16, 4, 16, 4, 8, 9, 12, 8, 12, 4, 20
Offset: 1

Views

Author

Vaclav Kotesovec, May 11 2024

Keywords

Comments

In general, for p prime, Sum_{j=1..n} tau(j*p) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.
If n is in A033428, then a(n) is odd and vice versa. - R. J. Mathar, Amiram Eldar, May 20 2024.

Links

Crossrefs

Cf. A000005, A001620, A033428, A099777, A372714, A372715, A372786, A372789, A372792.
Cf. also A144613.

Programs

  • Mathematica
    Table[DivisorSigma[0, 3*n], {n, 1, 150}]
  • PARI
    a(n) = numdiv(3*n); \\ Michel Marcus, May 20 2024

Formula

Sum_{k=1..n} a(k) ~ n * (5*(log(n) + 2*gamma - 1) + log(3)) / 3, where gamma is the Euler-Mascheroni constant A001620.

A372784 a(n) = tau(4*n) = A000005(4*n).

Original entry on oeis.org

3, 4, 6, 5, 6, 8, 6, 6, 9, 8, 6, 10, 6, 8, 12, 7, 6, 12, 6, 10, 12, 8, 6, 12, 9, 8, 12, 10, 6, 16, 6, 8, 12, 8, 12, 15, 6, 8, 12, 12, 6, 16, 6, 10, 18, 8, 6, 14, 9, 12, 12, 10, 6, 16, 12, 12, 12, 8, 6, 20, 6, 8, 18, 9, 12, 16, 6, 10, 12, 16, 6, 18, 6, 8, 18, 10
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A193553, A366872.
Cf. A000005, A099777, A372713, A372785, A372786, A372787, A372788, A372789, A372790, A372791, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 4*n], {n, 1, 150}]

Formula

For n > 1, a(n) = A366872(n-2).
Sum_{k=1..n} a(k) ~ (8*n*(log(n) + 2*gamma - 1) + n*4*log(2)) / 4, where gamma is the Euler-Mascheroni constant A001620.

A372786 Number of divisors of 6n; a(n) = tau(6*n) = A000005(6*n).

Original entry on oeis.org

4, 6, 6, 8, 8, 9, 8, 10, 8, 12, 8, 12, 8, 12, 12, 12, 8, 12, 8, 16, 12, 12, 8, 15, 12, 12, 10, 16, 8, 18, 8, 14, 12, 12, 16, 16, 8, 12, 12, 20, 8, 18, 8, 16, 16, 12, 8, 18, 12, 18, 12, 16, 8, 15, 16, 20, 12, 12, 8, 24, 8, 12, 16, 16, 16, 18, 8, 16, 12, 24, 8, 20
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A224613.
Cf. A000005, A001620, A099777, A372713, A372784, A372785, A372787, A372788, A372789, A372790, A372791, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 6*n], {n, 1, 150}]
  • PARI
    A372786(n) = numdiv(6*n); \\ Antti Karttunen, Jul 19 2024

Formula

Sum_{k=1..n} a(k) ~ (15*n*(log(n) + 2*gamma - 1) + n*(5*log(2) + 3*log(3))) / 6, where gamma is the Euler-Mascheroni constant A001620.

A372792 Number of divisors of 12n; a(n) = tau(12*n) = A000005(12*n).

Original entry on oeis.org

6, 8, 9, 10, 12, 12, 12, 12, 12, 16, 12, 15, 12, 16, 18, 14, 12, 16, 12, 20, 18, 16, 12, 18, 18, 16, 15, 20, 12, 24, 12, 16, 18, 16, 24, 20, 12, 16, 18, 24, 12, 24, 12, 20, 24, 16, 12, 21, 18, 24, 18, 20, 12, 20, 24, 24, 18, 16, 12, 30, 12, 16, 24, 18, 24, 24
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Comments

In general, for m>=1, Sum_{j=1..n} tau(m*j) = A018804(m) * n * log(n) + O(n).
If p is prime, then Sum_{j=1..n} tau(p*j) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

Links

Crossrefs

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372788, A372789, A372790, A372791.

Programs

  • Mathematica
    Table[DivisorSigma[0, 12*n], {n, 1, 150}]
  • PARI
    A372792(n) = numdiv(12*n); \\ Antti Karttunen, Jul 19 2024

Formula

Sum_{k=1..n} a(k) ~ (40*n*(log(n) + 2*gamma - 1) + n*(20*log(2) + 8*log(3))) / 12, where gamma is the Euler-Mascheroni constant A001620.

A372785 a(n) = tau(5*n) = A000005(5*n).

Original entry on oeis.org

2, 4, 4, 6, 3, 8, 4, 8, 6, 6, 4, 12, 4, 8, 6, 10, 4, 12, 4, 9, 8, 8, 4, 16, 4, 8, 8, 12, 4, 12, 4, 12, 8, 8, 6, 18, 4, 8, 8, 12, 4, 16, 4, 12, 9, 8, 4, 20, 6, 8, 8, 12, 4, 16, 6, 16, 8, 8, 4, 18, 4, 8, 12, 14, 6, 16, 4, 12, 8, 12, 4, 24, 4, 8, 8, 12, 8, 16, 4, 15, 10, 8, 4, 24, 6, 8, 8, 16, 4, 18, 8, 12, 8, 8, 6
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A283118.
Cf. A000005, A099777, A372713, A372784, A372786, A372787, A372788, A372789, A372790, A372791, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 5*n], {n, 1, 150}]
  • PARI
    A372785(n) = numdiv(5*n); \\ Antti Karttunen, Jan 13 2025

Formula

Sum_{k=1..n} a(k) ~ (9*n*(log(n) + 2*gamma - 1) + n*log(5)) / 5, where gamma is the Euler-Mascheroni constant A001620.

Extensions

More terms from Antti Karttunen, Jan 13 2025

A372787 a(n) = tau(7*n) = A000005(7*n).

Original entry on oeis.org

2, 4, 4, 6, 4, 8, 3, 8, 6, 8, 4, 12, 4, 6, 8, 10, 4, 12, 4, 12, 6, 8, 4, 16, 6, 8, 8, 9, 4, 16, 4, 12, 8, 8, 6, 18, 4, 8, 8, 16, 4, 12, 4, 12, 12, 8, 4, 20, 4, 12, 8, 12, 4, 16, 8, 12, 8, 8, 4, 24, 4, 8, 9, 14, 8, 16, 4, 12, 8, 12, 4, 24, 4, 8, 12, 12, 6, 16, 4
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A283078.
Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372788, A372789, A372790, A372791, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 7*n], {n, 1, 150}]

Formula

Sum_{k=1..n} a(k) ~ (13*n*(log(n) + 2*gamma - 1) + n*log(7)) / 7, where gamma is the Euler-Mascheroni constant A001620.

A372788 a(n) = tau(8*n) = A000005(8*n).

Original entry on oeis.org

4, 5, 8, 6, 8, 10, 8, 7, 12, 10, 8, 12, 8, 10, 16, 8, 8, 15, 8, 12, 16, 10, 8, 14, 12, 10, 16, 12, 8, 20, 8, 9, 16, 10, 16, 18, 8, 10, 16, 14, 8, 20, 8, 12, 24, 10, 8, 16, 12, 15, 16, 12, 8, 20, 16, 14, 16, 10, 8, 24, 8, 10, 24, 10, 16, 20, 8, 12, 16, 20, 8, 21
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A283122.
Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372789, A372790, A372791, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 8*n], {n, 1, 150}]

Formula

Sum_{k=1..n} a(k) ~ (20*n*(log(n) + 2*gamma - 1) + n*12*log(2)) / 8, where gamma is the Euler-Mascheroni constant A001620.

A372790 a(n) = tau(10*n) = A000005(10*n).

Original entry on oeis.org

4, 6, 8, 8, 6, 12, 8, 10, 12, 9, 8, 16, 8, 12, 12, 12, 8, 18, 8, 12, 16, 12, 8, 20, 8, 12, 16, 16, 8, 18, 8, 14, 16, 12, 12, 24, 8, 12, 16, 15, 8, 24, 8, 16, 18, 12, 8, 24, 12, 12, 16, 16, 8, 24, 12, 20, 16, 12, 8, 24, 8, 12, 24, 16, 12, 24, 8, 16, 16, 18, 8, 30
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372788, A372789, A372791, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 10*n], {n, 1, 150}]

Formula

Sum_{k=1..n} a(k) ~ (27*n*(log(n) + 2*gamma - 1) + n*(9*log(2) + 3*log(5))) / 10, where gamma is the Euler-Mascheroni constant A001620.

A372791 a(n) = tau(11*n) = A000005(11*n).

Original entry on oeis.org

2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 3, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 6, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 6, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 9, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 6, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 12, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 6, 16
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Links

Crossrefs

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372788, A372789, A372790, A372792.

Programs

  • Mathematica
    Table[DivisorSigma[0, 11*n], {n, 1, 150}]

Formula

Sum_{k=1..n} a(k) ~ (21*n*(log(n) + 2*gamma - 1) + n*log(11)) / 11, where gamma is the Euler-Mascheroni constant A001620.

A372793 Sequence related to the asymptotic expansion of Sum_{k=1..n} tau(m*k).

Original entry on oeis.org

1, 2, 3, 16, 5, 864, 7, 4096, 729, 64000, 11, 6879707136, 13, 2809856, 61509375, 4294967296, 17, 812479653347328, 19, 26843545600000000, 26795786661, 2791309312, 23, 4019988717840603673710821376, 9765625, 73719087104, 7625597484987, 25962355635465062711296, 29
Offset: 1

Views

Author

Vaclav Kotesovec, May 13 2024

Keywords

Comments

For m>=1, Sum_{k=1..n} tau(m*k) = A018804(m) * n * log(n) + O(n).
If p is prime, then Sum_{k=1..n} tau(p*k) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

Examples

			Sum_{k=1..n} tau(4*k) ~ (8*n*(log(n) + 2*gamma - 1) + n*4*log(2)) / 4, a(4) = exp(4*log(2)) = 16.
Sum_{k=1..n} tau(6*k) ~ (15*n*(log(n) + 2*gamma - 1) + n*(5*log(2) + 3*log(3))) / 6, a(6) = exp(5*log(2) + 3*log(3)) = 864.
Sum_{k=1..n} tau(8*k) ~ (20*n*(log(n) + 2*gamma - 1) + n*12*log(2)) / 8, a(8) = exp(12*log(2)) = 4096.
Sum_{k=1..n} tau(9*k) ~ (21*n*(log(n) + 2*gamma - 1) + n*6*log(3)) / 9, a(9) = exp(6*log(3)) = 729.
Sum_{k=1..n} tau(10*k) ~ (27*n*(log(n) + 2*gamma - 1) + n*(9*log(2) + 3*log(5))) / 10, a(10) = exp(9*log(2) + 3*log(5)) = 64000.
Sum_{k=1..n} tau(12*k) ~ (40*n*(log(n) + 2*gamma - 1) + n*(20*log(2) + 8*log(3))) / 12, a(12) = exp(20*log(2) + 8*log(3)) = 6879707136.

Crossrefs

Cf. A000005 (m=1), A099777 (m=2), A372713 (m=3), A372784 (m=4), A372785 (m=5), A372786 (m=6), A372787 (m=7), A372788 (m=8), A372789 (m=9), A372790 (m=10), A372791 (m=11), A372792 (m=12).
Cf. A018804, A193679.

Formula

Sum_{k=1..n} tau(m*k) ~ A018804(m) * n * (log(n) - 1 + 2*gamma)/m + n*log(a(m))/m.
a(m) = exp(limit_{n->oo} (m * (Sum_{k=1..n} tau(m*k)) - A018804(m)*n*(log(n) - 1 + 2*gamma))/n).
If p is prime, then a(p) = p.
If p is prime, then a(p^k) = p^(k*p^(k-1)).
If p and q are distinct primes, then a(p*q) = p^(2*q-1) * q^(2*p-1).
Showing 1-10 of 10 results.

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