A034761 -id:A034761 - cp's OEIS frontend

A064840 a(n) = tau(n)*sigma(n).

1, 6, 8, 21, 12, 48, 16, 60, 39, 72, 24, 168, 28, 96, 96, 155, 36, 234, 40, 252, 128, 144, 48, 480, 93, 168, 160, 336, 60, 576, 64, 378, 192, 216, 192, 819, 76, 240, 224, 720, 84, 768, 88, 504, 468, 288, 96, 1240, 171, 558, 288, 588, 108, 960, 288, 960, 320, 360

Offset: 1

Vladeta Jovovic, Oct 25 2001

Dirichlet convolution of A034761 with (the Dirichlet inverse of A037213). - R. J. Mathar, Feb 11 2011

			For n = 10, a(10) = sigma(10) * tau(10) = 18 * 4 = 72. - _Indranil Ghosh_, Jan 20 2017

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A000005, A000203, A007503, A062354, A062355.

[ NumberOfDivisors(n)*SumOfDivisors(n) : n in [1..40]];

with(numtheory): seq(sigma(n)*tau(n), n=1..58) ; # Zerinvary Lajos, Jun 04 2008

Table[ DivisorSigma[0, n] * DivisorSigma[1, n], {n, 1, 58}] (* Jean-François Alcover, Mar 26 2013 *)

{ for (n=1, 1000, a=numdiv(n)*sigma(n); write("b064840.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

Multiplicative with a(p^e) = (p^(e+1)-1)*(e+1)/(p-1). a(n) = (1/2)*Sum_{i|n, j|n} (i+j).
Dirichlet g.f. (zeta(s)*zeta(s-1))^2/zeta(2s-1). - R. J. Mathar, Feb 11 2011
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (144*Zeta(3)) * (2*log(n) - 1 + 4*gamma - 4*Zeta'(3)/Zeta(3) + 24*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

A134577 A127170 * A127648.

Original entry on oeis.org

1, 2, 2, 2, 0, 3, 3, 4, 0, 4, 2, 0, 0, 0, 5, 4, 4, 6, 0, 0, 6, 2, 0, 0, 0, 0, 0, 7, 4, 6, 0, 8, 0, 0, 0, 8, 3, 0, 6, 0, 0, 0, 0, 0, 9, 4, 4, 0, 0, 10, 0, 0, 0, 0, 10

Offset: 1

Gary W. Adamson, Nov 02 2007

Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, 26, ...).
Left border = A000005: (1, 2, 2, 3, 2, 4, 2, ...).
A134577 * [1/1, 1/2, 1/3, ...] = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).
A134577 * [1, 2, 3, ...] = A007433: (1, 6, 11, 27, 27, 66, ...).
A134577 * A000005 = A034761: (1, 6, 8, 23, 12, 48, ...).

			First few rows of the triangle:
  1;
  2, 2;
  2, 0, 3;
  3, 4, 0, 4;
  2, 0, 0, 0, 5
  4, 4, 6, 0, 0, 6;
  2, 0, 0, 0, 0, 0, 7;
  4, 6, 0, 8, 0, 0, 0, 8;
  ...

Cf. A051731, A127170, A127648, A127093, A007429, A127170, A007433, A034761.

A127170 * A127648 = A051731^2 * A127648 = A051731 * A127093.

A328487 Dirichlet g.f.: zeta(s)^2 * zeta(s-1)^2 * (1 - 2^(1 - s))^2.

Original entry on oeis.org

1, 2, 8, 3, 12, 16, 16, 4, 42, 24, 24, 24, 28, 32, 96, 5, 36, 84, 40, 36, 128, 48, 48, 32, 98, 56, 184, 48, 60, 192, 64, 6, 192, 72, 192, 126, 76, 80, 224, 48, 84, 256, 88, 72, 504, 96, 96, 40, 178, 196, 288, 84, 108, 368, 288, 64, 320, 120, 120, 288, 124, 128, 672, 7, 336

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Dirichlet convolution of A000593 with itself.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000593, A001620, A034761, A328486.

nmax = 65; A000593 = Table[DivisorSum[n, Mod[#, 2] # &], {n, 1, nmax}]; Table[DivisorSum[n, A000593[[#]] A000593[[n/#]] &], {n, 1, nmax}]
f[p_, e_] := ((e+1)*p^(e+3) - (e+3)*(p^(e+2) - p + 1) + 2)/(p-1)^3; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

a(n) = Sum_{d|n} A000593(d) * A000593(n/d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (Pi^2 * (log(n)/2 + log(2) + gamma - 1/4) + 6*zeta'(2)) / 144, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(2^e) = e+1, and a(p^e) = ((e+1)*p^(e+3) - (e+3)*(p^(e+2) - p + 1) + 2)/(p-1)^3 for an odd prime p. - Amiram Eldar, Sep 15 2023

A349711 a(n) = Sum_{d|n} sopfr(d) * sopfr(n/d).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 16, 9, 20, 0, 44, 0, 28, 30, 40, 0, 54, 0, 68, 42, 44, 0, 104, 25, 52, 36, 92, 0, 124, 0, 80, 66, 68, 70, 147, 0, 76, 78, 152, 0, 164, 0, 140, 108, 92, 0, 200, 49, 110, 102, 164, 0, 144, 110, 200, 114, 116, 0, 298, 0, 124, 144, 140, 130, 244, 0, 212, 138, 236, 0, 300, 0, 148, 140

Offset: 1

Ilya Gutkovskiy, Nov 26 2021

Dirichlet convolution of A001414 with itself.

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

Cf. A000292, A001414, A034761, A318366, A349712.

b:= proc(n) option remember; add(i[1]*i[2], i=ifactors(n)[2]) end:
a:= n-> add(b(d)*b(n/d), d=numtheory[divisors](n)):
seq(a(n), n=1..75);  # Alois P. Heinz, Nov 26 2021

sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger@n; a[n_] := Sum[sopfr[d] sopfr[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]

sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
a(n) = sumdiv(n, d, sopfr(d)*sopfr(n/d)); \\ Michel Marcus, Nov 26 2021

from itertools import product
from sympy import factorint
def A349711(n):
    f = factorint(n)
    plist, m = list(f.keys()), sum(f[p]*p for p in f)
    return sum((lambda x: x*(m-x))(sum(d[i]*p for i, p in enumerate(plist))) for d in product(*(list(range(f[p]+1)) for p in plist))) # Chai Wah Wu, Nov 27 2021

Dirichlet g.f.: ( zeta(s) * Sum_{p prime} p/(p^s-1) )^2.

a(p^k) = (k^3-k)*p^2/6 = A000292(k-1)*p^2 for p prime. - Chai Wah Wu, Nov 28 2021

A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 32, 0, 28, 30, 12, 0, 42, 0, 48, 42, 44, 0, 52, 25, 52, 18, 64, 0, 124, 0, 16, 66, 68, 70, 87, 0, 76, 78, 76, 0, 164, 0, 96, 78, 92, 0, 72, 49, 90, 102, 112, 0, 72, 110, 100, 114, 116, 0, 234, 0, 124, 102, 20, 130, 244, 0, 144, 138, 236, 0, 132, 0, 148, 110

Offset: 1

Ilya Gutkovskiy, Nov 26 2021

nonn

Dirichlet convolution of A008472 with itself.

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

Cf. A008472, A034761, A061397, A070288, A319131, A349711.

sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]

sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ Michel Marcus, Nov 26 2021

from sympy import divisors, factorint
def sopf(n): return sum(factorint(n))
def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Nov 26 2021

Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.
a(n) = Sum_{d|n} A061397(d) * A319131(n/d).
a(p) = 0 for p prime. - Michael S. Branicky, Nov 26 2021
a(p^k) = (k-1)*p^2 for p prime and k > 0. - Chai Wah Wu, Nov 28 2021

A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

Original entry on oeis.org

1, 10, 20, 67, 52, 200, 100, 380, 282, 520, 244, 1340, 340, 1000, 1040, 1973, 580, 2820, 724, 3484, 2000, 2440, 1060, 7600, 1978, 3400, 3460, 6700, 1684, 10400, 1924, 9710, 4880, 5800, 5200, 18894, 2740, 7240, 6800, 19760, 3364, 20000, 3700, 16348, 14664

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Dirichlet convolution of A001157 with itself.
Dirichlet convolution of A000005 with A034714.
Dirichlet convolution of A000290 with A007433.

Cf. A000005, A000290, A001157, A001620, A007426, A007433, A034714, A034761.

Cf. A175705.

[&+[DivisorSigma(2,d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Oct 16 2019

Table[DivisorSum[n, DivisorSigma[2, #] DivisorSigma[2, n/#] &], {n, 1, 45}]
f[p_, e_] :=((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3 ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 / (1 - p^2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

a(n) = Sum_{d|n} sigma_2(d) * sigma_2(n/d), where sigma_2 = A001157.
a(n) = Sum_{d|n} d^2 * tau(d) * tau(n/d), where tau = A000005.
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 * (zeta(3)*(log(n)/3 + 2*gamma/3 - 1/9) + 2*zeta'(3)/3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = ((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3. - Amiram Eldar, Sep 15 2023

A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

Original entry on oeis.org

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

A343569 If n = Product (p_j^k_j) then a(n) = Product (2*(p_j^k_j + 1)), with a(1) = 1.

Original entry on oeis.org

1, 6, 8, 10, 12, 48, 16, 18, 20, 72, 24, 80, 28, 96, 96, 34, 36, 120, 40, 120, 128, 144, 48, 144, 52, 168, 56, 160, 60, 576, 64, 66, 192, 216, 192, 200, 76, 240, 224, 216, 84, 768, 88, 240, 240, 288, 96, 272, 100, 312, 288, 280, 108, 336, 288, 288, 320, 360, 120, 960, 124, 384, 320, 130

Offset: 1

Ilya Gutkovskiy, Apr 20 2021

Cf. A001221, A034444, A034448, A034761, A064840, A107759, A333557, A343525.

a[1] = 1; a[n_] := Times @@ (2 (#[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 64}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2] + 2; f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 20 2021

a(n) = usigma(n) * 2^omega(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d) * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d) * 2^omega(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A343525(d).

A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

Original entry on oeis.org

1, 6, 8, 19, 12, 48, 16, 48, 36, 72, 24, 152, 28, 96, 96, 113, 36, 216, 40, 228, 128, 144, 48, 384, 88, 168, 136, 304, 60, 576, 64, 258, 192, 216, 192, 684, 76, 240, 224, 576, 84, 768, 88, 456, 432, 288, 96, 904, 164, 528, 288, 532, 108, 816, 288, 768, 320, 360, 120, 1824

Offset: 1

Ilya Gutkovskiy, Nov 29 2021

Dirichlet convolution of A034448 with itself.

Cf. A034448, A034761, A322328, A328485, A343569.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a[n_] := Sum[usigma[d] usigma[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

Dirichlet g.f.: ( zeta(s) * zeta(s-1) / zeta(2*s-1) )^2.
Multiplicative with a(p^e) = e * (p^e + 1) + (p+1) * (p^e - 1)/(p-1). - Amiram Eldar, Nov 29 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / zeta(3)^2 * (Pi^2 * log(n)/72 + gamma * Pi^2/36 - Pi^2/144 + zeta'(2)/6 - Pi^2 * zeta'(3)/(18*zeta(3))), where zeta(3) = A002117, zeta'(2) = -A073002, zeta'(3) = -A244115 and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2021

cp's OEIS Frontend

A064840 a(n) = tau(n)*sigma(n).

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A134577 A127170 * A127648.

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A328487 Dirichlet g.f.: zeta(s)^2 * zeta(s-1)^2 * (1 - 2^(1 - s))^2.

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A349711 a(n) = Sum_{d|n} sopfr(d) * sopfr(n/d).

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A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

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A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

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A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

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A343569 If n = Product (p_j^k_j) then a(n) = Product (2*(p_j^k_j + 1)), with a(1) = 1.