A035116 -id:A035116 - cp's OEIS frontend

A049645 Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.

1, 3, 7, 11, 19, 21, 23, 31, 33, 35, 43, 47, 57, 59, 62, 67, 69, 71, 77, 79, 83, 91, 93, 94, 103, 105, 107, 115, 119, 127, 129, 131, 133, 139, 141, 151, 155, 158, 161, 163, 167, 177, 179, 186, 189, 191, 199, 201, 203, 209, 211, 213, 217, 223

Offset: 1

N. J. A. Sloane

nonn

Bateman et al. (1981) proved that the asymptotic density of this sequence is 1/2. - Amiram Eldar, Jan 16 2020

Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, section 51, page 119.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Paul T. Bateman, Paul Erdős, Carl Pomerance and E.G. Straus, The arithmetic mean of the divisors of an integer, in Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer, Berlin - New York, 1981, pp. 197-220, alternative link.

Cf. A003601, A049642, A049692.

Cf. A000203, A035116.

[k:k in [1..230]| DivisorSigma(1,k) mod (DivisorSigma(0,k))^2 eq 0]; // Marius A. Burtea, Jan 16 2020

with(numtheory): t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod sigma[ 0 ](n)^2 = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t;

Select[Range[1, 250], Mod[DivisorSigma[1, #], DivisorSigma[0, #]^2] == 0 &] (* G. C. Greubel, Dec 06 2017 *)

isok(n) = sigma(n) % numdiv(n)^2 == 0; \\ Michel Marcus, Dec 07 2017

{n: A035116(n) | A000203(n)}. - R. J. Mathar, Jan 29 2019

A110601 a(n) = phi(n)*tau(n)^2, where phi is Euler's totient function and tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 4, 8, 18, 16, 32, 24, 64, 54, 64, 40, 144, 48, 96, 128, 200, 64, 216, 72, 288, 192, 160, 88, 512, 180, 192, 288, 432, 112, 512, 120, 576, 320, 256, 384, 972, 144, 288, 384, 1024, 160, 768, 168, 720, 864, 352, 184, 1600, 378, 720, 512, 864, 208, 1152, 640

Offset: 1

Emeric Deutsch, Jul 29 2005

			a(4)=18 because phi(4)=2 and tau(4)=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Stefan Porubsky and M. G. Greening, Problem E2351, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 436.

Cf. A000005, A000010, A035116, A062355.

[EulerPhi(n)*NumberOfDivisors(n)^2: n in [1..60]]; // Vincenzo Librandi, Jun 21 2017

with(numtheory): a:=n->phi(n)*tau(n)^2: seq(a(n),n=1..70);

Table[EulerPhi[n]DivisorSigma[0,n]^2,{n,60}] (* Harvey P. Dale, Nov 29 2011 *)

a(n) = eulerphi(n)*numdiv(n)^2; \\ Michel Marcus, Jun 21 2017

a(n) = A000010(n) * A035116(n) = A062355(n) * A000005(n). - R. J. Mathar, Jul 26 2022

Multiplicative with a(p^e) = (e+1)^2*(p-1)*p^(e-1). - Amiram Eldar, Dec 29 2022

A162664 a(n) = sigma(n) + tau(n)^2.

Original entry on oeis.org

2, 7, 8, 16, 10, 28, 12, 31, 22, 34, 16, 64, 18, 40, 40, 56, 22, 75, 24, 78, 48, 52, 28, 124, 40, 58, 56, 92, 34, 136, 36, 99, 64, 70, 64, 172, 42, 76, 72, 154, 46, 160, 48, 120, 114, 88, 52, 224, 66, 129, 88, 134, 58, 184, 88, 184, 96, 106, 64, 312, 66, 112, 140, 176, 100

Offset: 1

Franklin T. Adams-Watters, Jul 09 2009

nonn

Row 2 of A162663.

Shawn A. Broyles, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A035116, A162663.

Table[DivisorSigma[1, n] + DivisorSigma[0, n]^2, {n, 65}] (* Robert A. Russell, Apr 28 2018 *)

a(n) = sigma(n) + numdiv(n)^2; \\ Michel Marcus, Apr 28 2018

A272209 Number of partitions of the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 11, 2, 5, 5, 7, 2, 11, 2, 11, 5, 5, 2, 22, 3, 5, 5, 11, 2, 22, 2, 11, 5, 5, 5, 30, 2, 5, 5, 22, 2, 22, 2, 11, 11, 5, 2, 42, 3, 11, 5, 11, 2, 22, 5, 22, 5, 5, 2, 77, 2, 5, 11, 15, 5, 22, 2, 11, 5, 22, 2, 77, 2, 5, 11, 11, 5, 22, 2, 42, 7, 5, 2, 77

Offset: 1

Omar E. Pol, Apr 25 2016

nonn

			For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12. There are 6 divisors of 12 and the number of partitions of 6 is A000041(6) = 11, so a(12) = 11.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000041, A035116, A058699, A085543, A272024.

Table[PartitionsP@ DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = numbpart(numdiv(n)); \\ Michel Marcus, Apr 26 2016

a(n) = p(d(n)) = A000041(A000005(n)).

A301746 Expansion of Product_{k>=1} (1 + x^k)^(sigma_0(k)^2).

Original entry on oeis.org

1, 1, 4, 8, 19, 35, 82, 142, 291, 524, 989, 1724, 3174, 5393, 9517, 16064, 27464, 45481, 76357, 124402, 204497, 329559, 532316, 846564, 1349481, 2120814, 3335819, 5191522, 8070062, 12434176, 19136484, 29215324, 44531151, 67431985, 101882975, 153055897

Offset: 0

Vaclav Kotesovec, Mar 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A035116, A107742, A301747.

nmax = 50; CoefficientList[Series[Product[(1+x^k)^(DivisorSigma[0, k]^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]^2, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 29 2018 *)

Conjecture: log(a(n)) ~ sqrt(n) * log(n)^(3/2) / (2*sqrt(6)). - Vaclav Kotesovec, Aug 29 2018

A301747 Expansion of Product_{k>=1} (1/(1 - x^k))^(sigma_0(k)^2).

Original entry on oeis.org

1, 1, 5, 9, 28, 48, 130, 226, 532, 941, 2021, 3545, 7210, 12509, 24209, 41715, 77742, 132404, 239655, 403731, 712426, 1188079, 2052070, 3386854, 5745200, 9388740, 15672560, 25376167, 41765597, 67021171, 108932532, 173327693, 278533669, 439653317, 699265665

Offset: 0

Vaclav Kotesovec, Mar 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A006171, A035116, A301746.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^(DivisorSigma[0, k]^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k]^2, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 29 2018 *)

log(a(n)) ~ sqrt(n) * log(n)^(3/2) / (2*sqrt(3)). - Vaclav Kotesovec, Aug 28 2018

A319089 a(n) = tau(n)^3, where tau is A000005.

Original entry on oeis.org

1, 8, 8, 27, 8, 64, 8, 64, 27, 64, 8, 216, 8, 64, 64, 125, 8, 216, 8, 216, 64, 64, 8, 512, 27, 64, 64, 216, 8, 512, 8, 216, 64, 64, 64, 729, 8, 64, 64, 512, 8, 512, 8, 216, 216, 64, 8, 1000, 27, 216, 64, 216, 8, 512, 64, 512, 64, 64, 8, 1728, 8, 64, 216, 343

Offset: 1

Vaclav Kotesovec, Sep 10 2018

Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A006218, A035116, A061502, A318755 (partial sums).

with(numtheory): seq(tau(n)^3, n=1..100); # Ridouane Oudra, Mar 07 2023

Mathematica
```
DivisorSigma[0, Range[100]]^3
```

a(n) = numdiv(n)^3; \\ Altug Alkan, Sep 10 2018

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X + X^2)/(1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Mar 09 2023

Multiplicative with a(p^e) = (e+1)^3. - Amiram Eldar, Dec 31 2022
a(n) = Sum_{d1|n} Sum_{d2|n} tau(d1*d2). - Ridouane Oudra, Mar 07 2023
From Vaclav Kotesovec, Mar 09 2023: (Start)
Dirichlet g.f.: Product_{p prime} p^(2*s) * (1 + 4*p^s + p^(2*s)) / (p^s - 1)^4.
Dirichlet g.f.: zeta(s)^8 * Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), (with a product that converges for s=1). (End)

A351521 Dirichlet g.f.: Product_{p prime} (1 + 4*p^(-s)).

Original entry on oeis.org

1, 4, 4, 0, 4, 16, 4, 0, 0, 16, 4, 0, 4, 16, 16, 0, 4, 0, 4, 0, 16, 16, 4, 0, 0, 16, 0, 0, 4, 64, 4, 0, 16, 16, 16, 0, 4, 16, 16, 0, 4, 64, 4, 0, 0, 16, 4, 0, 0, 0, 16, 0, 4, 0, 16, 0, 16, 16, 4, 0, 4, 16, 0, 0, 16, 64, 4, 0, 16, 64, 4, 0, 4, 16, 0, 0, 16, 64

Offset: 1

Vaclav Kotesovec, Feb 17 2022

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A074823, A347149.
Cf. A061142, A165824, A165825.
Cf. A008966, A035116.

Table[MoebiusMu[n]^2 * 4^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X))[n], ", "))

Dirichlet g.f.: zeta(s)^4 * Product_{prime p} (1 + (4 - 15*p^s + 20*p^(2*s) - 10*p^(3*s))/p^(5*s)).
a(n) = A008966(n) * A035116(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 4, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

A361147 a(n) = sigma(n)^3.

Original entry on oeis.org

1, 27, 64, 343, 216, 1728, 512, 3375, 2197, 5832, 1728, 21952, 2744, 13824, 13824, 29791, 5832, 59319, 8000, 74088, 32768, 46656, 13824, 216000, 29791, 74088, 64000, 175616, 27000, 373248, 32768, 250047, 110592, 157464, 110592, 753571, 54872, 216000, 175616

Offset: 1

Vaclav Kotesovec, Mar 02 2023

Cf. A000203, A024916, A072861, A072379, A361179.

Cf. A000005, A000578, A035116, A319089.

Mathematica
```
Table[DivisorSigma[1, n]^3, {n, 1, 50}]
```
PARI
```
a(n) = sigma(n)^3;
```

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X*(2 + 2*p + p^2*X)) / ((1-X)*(1-p*X)*(1-p^2*X)*(1-p^3*X)))[n], ", "))

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^3.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * n^4 / 2160, where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.83598357433419286770442457158038489640898183...
a(n) = A000578(A000203(n)).

A065814 a(n) = tau(n)^2 - tau(n^2), where tau(n) = A000005(n).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 9, 4, 7, 1, 21, 1, 7, 7, 16, 1, 21, 1, 21, 7, 7, 1, 43, 4, 7, 9, 21, 1, 37, 1, 25, 7, 7, 7, 56, 1, 7, 7, 43, 1, 37, 1, 21, 21, 7, 1, 73, 4, 21, 7, 21, 1, 43, 7, 43, 7, 7, 1, 99, 1, 7, 21, 36, 7, 37, 1, 21, 7, 37, 1, 109, 1, 7, 21, 21, 7, 37, 1, 73, 16, 7, 1, 99, 7, 7, 7, 43

Offset: 1

Labos Elemer, Nov 22 2001

nonn

If n = p^c = power of a prime, then a(n) = (c+1)^2 - (2c+1) = c^2. If n is squarefree with k prime factors then a(n) = 4^k - 3^k, so A065814(A002110(n)) = 4^n - 3^n = A005061(n). Terms depend on prime signature only.

If n is a prime (A000040), then a(n) = 1. If n is a semiprime (A001358), then a(n) = 4 + 3*ceiling(sqrt(n)) - 3*floor(sqrt(n)). If n is a triprime (A014612), then a(n) = 9 * floor(1/omega(n)) + 21 * (1 - (omega(n) mod 2)) + 37 * floor(omega(n)/3), n > 1. - Wesley Ivan Hurt, May 24 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A000040, A000290, A000961, A001358, A002110, A005061, A014612, A035116, A048691.

a[n_] := DivisorSigma[0, n]^2 - DivisorSigma[0, n^2]; Array[a, 100] (* Amiram Eldar, Apr 25 2024 *)

a(n) = { numdiv(n)^2 - numdiv(n^2) } \\ Harry J. Smith, Oct 31 2009

a(n) = A000005(n)^2 - A000005(n^2).
G.f.: Sum_{n>=1} A000005(n^2)*x^(2*n)/(1-x^n). - Mircea Merca, Feb 26 2014
a(n) = A035116(n) - A048691(n). - Amiram Eldar, Apr 25 2024

cp's OEIS Frontend

A049645 Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.

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A110601 a(n) = phi(n)*tau(n)^2, where phi is Euler's totient function and tau(n) is the number of divisors of n.

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A162664 a(n) = sigma(n) + tau(n)^2.

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A272209 Number of partitions of the number of divisors of n.

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A301746 Expansion of Product_{k>=1} (1 + x^k)^(sigma_0(k)^2).

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A301747 Expansion of Product_{k>=1} (1/(1 - x^k))^(sigma_0(k)^2).

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A319089 a(n) = tau(n)^3, where tau is A000005.

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A351521 Dirichlet g.f.: Product_{p prime} (1 + 4*p^(-s)).