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.

Previous Showing 11-15 of 15 results.

A372674 a(n) = Sum_{j=1..n} Sum_{k=1..n} tau(j*k).

Original entry on oeis.org

1, 8, 23, 54, 89, 162, 221, 326, 439, 596, 707, 964, 1107, 1352, 1645, 1976, 2179, 2630, 2865, 3390, 3859, 4316, 4615, 5406, 5883, 6444, 7059, 7892, 8299, 9430, 9877, 10794, 11635, 12424, 13361, 14852, 15415, 16324, 17349, 18952, 19587, 21342, 22017, 23486, 25177
Offset: 1

Views

Author

Vaclav Kotesovec, May 10 2024

Keywords

Comments

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. A372633, A372675.
Cf. A006218, A263086.
cf. A000005, A099777, A372713.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[0, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[0, n^2] + 2*Sum[DivisorSigma[0, j*n], {j, 1, n - 1}], {n, 2, 50}]]

A372714 a(n) = tau(3*n-1) = A000005(3*n-1).

Original entry on oeis.org

2, 2, 4, 2, 4, 2, 6, 2, 4, 2, 6, 4, 4, 2, 6, 2, 6, 2, 8, 2, 4, 4, 6, 2, 4, 4, 10, 2, 4, 2, 6, 4, 6, 2, 8, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 2, 12, 4, 4, 2, 8, 4, 4, 4, 6, 2, 8, 2, 10, 2, 8, 4, 6, 2, 4, 2, 12, 4, 4, 4, 6, 4, 4, 4, 12, 2, 8, 2, 6, 2, 6, 6, 8, 2, 4, 2
Offset: 1

Views

Author

Vaclav Kotesovec, May 11 2024

Keywords

Crossrefs

Cf. A372713, A372715.
Cf. A000005, A099777, A099774.

Programs

  • Mathematica
    Table[DivisorSigma[0, 3*n-1], {n, 1, 150}]

Formula

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

A372715 a(n) = tau(3*n-2) = A000005(3*n-2).

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 2, 4, 3, 6, 2, 4, 2, 8, 2, 4, 3, 6, 4, 4, 2, 7, 2, 8, 2, 6, 2, 4, 4, 8, 4, 4, 2, 9, 2, 4, 2, 10, 4, 4, 3, 6, 2, 8, 4, 8, 2, 4, 4, 6, 2, 8, 2, 12, 2, 4, 3, 6, 6, 4, 2, 8, 4, 8, 2, 9, 2, 4, 4, 10, 2, 4, 4, 12, 2, 4, 2, 8, 4, 8, 2, 6, 4, 8, 4, 9
Offset: 1

Views

Author

Vaclav Kotesovec, May 11 2024

Keywords

Crossrefs

Cf. A372713, A372714.
Cf. A000005, A099777, A099774.

Programs

  • Mathematica
    Table[DivisorSigma[0, 3*n-2], {n, 1, 150}]

Formula

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

A328484 Dirichlet g.f.: zeta(s)^2 / (1 - 3^(-s)).

Original entry on oeis.org

1, 2, 3, 3, 2, 6, 2, 4, 6, 4, 2, 9, 2, 4, 6, 5, 2, 12, 2, 6, 6, 4, 2, 12, 3, 4, 10, 6, 2, 12, 2, 6, 6, 4, 4, 18, 2, 4, 6, 8, 2, 12, 2, 6, 12, 4, 2, 15, 3, 6, 6, 6, 2, 20, 4, 8, 6, 4, 2, 18, 2, 4, 12, 7, 4, 12, 2, 6, 6, 8, 2, 24, 2, 4, 9, 6, 4, 12, 2, 10, 15, 4, 2, 18, 4
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

Comments

Inverse Moebius transform of A051064.
Dirichlet convolution of A000005 with characteristic function of powers of 3.

Links

Crossrefs

Cf. A000005, A051064, A129628.
Cf. A007949, A372713, A373438, A035191.

Programs

  • Maple
    seq(add(padic[ordp](3*d, 3), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024
  • Mathematica
    Table[DivisorSum[n, IntegerExponent[3 #, 3] &], {n, 1, 85}]
nmax = 85; CoefficientList[Series[Sum[Sum[x^(i 3^j)/(1 - x^(i 3^j)), {j, 0, Floor[Log[3, nmax]] + 1}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 3, (e + 1)*(e + 2)/2, e + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Formula

G.f.: Sum_{i>=1} Sum_{j>=0} x^(i*3^j) / (1 - x^(i*3^j)).
a(n) = Sum_{d|n} A051064(d).
Sum_{k=1..n} a(k) ~ 3*n*(log(n)/2 - log(3)/4 - 1/2 + gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p=3, and e+1 otherwise. - Amiram Eldar, Dec 02 2020
From Ridouane Oudra, Sep 30 2024: (Start)
a(n) = Sum_{i=0..A007949(n)} tau(n/3^i).
a(n) = Sum_{d|3*n} A007949(d).
a(n) = (1/2)*A051064(n)*A372713(n).
a(n) = (1/2)*(A051064(n) + 1)*A000005(n).
a(n) = A373438(n)*A035191(n). (End)

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).
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.