A372784 -id:A372784 - cp's OEIS frontend

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

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

Vaclav Kotesovec, May 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A224613.

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

Table[DivisorSigma[0, 6*n], {n, 1, 150}]

A372786(n) = numdiv(6*n); \\ Antti Karttunen, Jul 19 2024

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.

A372789 Number of divisors of 9n; a(n) = tau(9n) = A000005(9n).

Original entry on oeis.org

3, 6, 4, 9, 6, 8, 6, 12, 5, 12, 6, 12, 6, 12, 8, 15, 6, 10, 6, 18, 8, 12, 6, 16, 9, 12, 6, 18, 6, 16, 6, 18, 8, 12, 12, 15, 6, 12, 8, 24, 6, 16, 6, 18, 10, 12, 6, 20, 9, 18, 8, 18, 6, 12, 12, 24, 8, 12, 6, 24, 6, 12, 10, 21, 12, 16, 6, 18, 8, 24, 6, 20, 6, 12, 12

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283123.

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372788, A372790, A372791, A372792.

Table[DivisorSigma[0, 9*n], {n, 1, 150}]

A372789(n) = numdiv(9*n); \\ Antti Karttunen, Jul 19 2024

Sum_{k=1..n} a(k) ~ (21*n*(log(n) + 2*gamma - 1) + n*6*log(3)) / 9, where gamma is the Euler-Mascheroni constant A001620.

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

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

Vaclav Kotesovec, May 13 2024

nonn

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.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

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

Table[DivisorSigma[0, 12*n], {n, 1, 150}]

A372792(n) = numdiv(12*n); \\ Antti Karttunen, Jul 19 2024

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(5n) = A000005(5n).

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

Vaclav Kotesovec, May 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283118.

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

Table[DivisorSigma[0, 5*n], {n, 1, 150}]

A372785(n) = numdiv(5*n); \\ Antti Karttunen, Jan 13 2025

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.

More terms from Antti Karttunen, Jan 13 2025

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

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

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283078.

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

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

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(8n) = A000005(8n).

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

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283122.

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

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

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(10n) = A000005(10n).

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

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

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

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

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(11n) = A000005(11n).

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

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

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

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

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.

A349693 Dirichlet convolution of the ruler function (A001511) with itself.

Original entry on oeis.org

1, 4, 2, 10, 2, 8, 2, 20, 3, 8, 2, 20, 2, 8, 4, 35, 2, 12, 2, 20, 4, 8, 2, 40, 3, 8, 4, 20, 2, 16, 2, 56, 4, 8, 4, 30, 2, 8, 4, 40, 2, 16, 2, 20, 6, 8, 2, 70, 3, 12, 4, 20, 2, 16, 4, 40, 4, 8, 2, 40, 2, 8, 6, 84, 4, 16, 2, 20, 4, 16, 2, 60, 2, 8, 6, 20, 4, 16, 2, 70

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Dirichlet convolution of A000005 with A104117. - Ridouane Oudra, Jul 23 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A001511, A115364, A129628.

Cf. A000005, A104117, A007814, A085058, A090739, A099777, A372784, A001227, A000292,

a:= n-> (f-> add(f(d)*f(n/d), d=numtheory[divisors](n)))(k-> padic[ordp](2*k, 2)):
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 25 2021

Table[Sum[IntegerExponent[2 d, 2] IntegerExponent[2 n/d, 2], {d, Divisors[n]}], {n, 1, 80}]
f[p_, e_] := If[p == 2, Binomial[e + 3, 3], e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 80] (* Amiram Eldar, Nov 25 2021 *)

A001511(n) = (1+valuation(n,2));
A349693(n) = sumdiv(n,d,A001511(n/d)*A001511(d)); \\ Antti Karttunen, Nov 25 2021

from sympy import divisor_count
def A349693(n): return divisor_count(n)*(m:=(n&-n).bit_length()+1)*(m+1)//6 # Chai Wah Wu, Jul 13 2022

Dirichlet g.f.: zeta(s)^2 * 4^s / (2^s-1)^2.
a(n) = Sum_{d|n} A001511(d) * A001511(n/d).
a(n) = Sum_{d|n} A000217(A001511(d)).
Multiplicative with a(p^e) = binomial(e+3,3) if p = 2 and e+1 otherwise. - Amiram Eldar, Nov 25 2021
Sum_{k=1..n} a(k) ~ 4*n*(log(n) - 1 + 2*gamma - 2*log(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 26 2021
From Ridouane Oudra, Jul 23 2025: (Start)
a(n) = Sum_{i=0..A007814(n)} (i+1)*tau(n/2^i).
a(n) = Sum_{d|n} A115364(d).
a(n) = (1/6)*A090739(n)*A085058(n-1)*A000005(n).
a(n) = (1/6)*A001511(n)*A090739(n)*A099777(n).
a(n) = (1/3)*A115364(n)*A372784(n).
a(n) = A001227(n)*A000292(A001511(n)).
a(2*n+1) = tau(2*n+1).
a(2^k*(2*n+1)) = binomial(k+3, 3)*tau(2*n+1), for k, n >= 0. (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

Vaclav Kotesovec, May 13 2024

nonn

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.

			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.

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.

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).

cp's OEIS Frontend

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

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A372789 Number of divisors of 9n; a(n) = tau(9*n) = A000005(9*n).

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

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

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

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

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

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

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A349693 Dirichlet convolution of the ruler function (A001511) with itself.

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

A372789 Number of divisors of 9n; a(n) = tau(9n) = A000005(9n).

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

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

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

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

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

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