A038040 -id:A038040 - cp's OEIS frontend

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

1, 1, 5, 11, 33, 67, 180, 366, 871, 1782, 3927, 7885, 16637, 32763, 66469, 128938, 253871, 484034, 930959, 1747304, 3292730, 6092664, 11282364, 20596790, 37568653, 67736175, 121886533, 217261372, 386216073, 681119439, 1197524035, 2091091902, 3639519280

Offset: 0

Vaclav Kotesovec, Jan 05 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A006171, A038040, A061256, A107742, A192065, A280541.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[k*DivisorSigma[0, k], 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 27 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

log(a(n)) ~ (3/2)^(2/3) * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3). - Vaclav Kotesovec, Aug 28 2018

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).

Original entry on oeis.org

1, 1, 4, 10, 24, 52, 125, 253, 549, 1126, 2290, 4525, 8987, 17259, 33174, 62669, 117425, 217295, 399904, 726984, 1314257, 2354807, 4191671, 7405590, 13009916, 22696115, 39384232, 67937488, 116584833, 199001304, 338076500, 571507377, 961855945, 1611567819

Offset: 0

Vaclav Kotesovec, Jan 05 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A006171, A038040, A061256, A107742, A192065, A280540.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[k*DivisorSigma[0, k], 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 27 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

Conjecture: log(a(n)) ~ 3 * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Aug 29 2018

A328259 a(n) = n * sigma_2(n).

Original entry on oeis.org

1, 10, 30, 84, 130, 300, 350, 680, 819, 1300, 1342, 2520, 2210, 3500, 3900, 5456, 4930, 8190, 6878, 10920, 10500, 13420, 12190, 20400, 16275, 22100, 22140, 29400, 24418, 39000, 29822, 43680, 40260, 49300, 45500, 68796, 50690, 68780, 66300, 88400, 68962, 105000, 79550, 112728, 106470

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

Moebius transform of A027847.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A001157, A027847, A038040, A064987, A275585, A281372, A294362.

Table[n DivisorSigma[2, n], {n, 1, 45}]
nmax = 45; CoefficientList[Series[Sum[k^3 x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = n*sigma(n, 2); \\ Michel Marcus, Dec 02 2020

G.f.: Sum_{k>=1} k^3 * x^k / (1 - x^k)^2.
G.f.: Sum_{k>=1} k * x^k * (1 + 4 * x^k + x^(2*k)) / (1 - x^k)^4.
Dirichlet g.f.: zeta(s - 1) * zeta(s - 3).
Sum_{k=1..n} a(k) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Oct 09 2019
Multiplicative with a(p^e) = (p^(3*e+2) - p^e)/(p^2 - 1). - Amiram Eldar, Dec 02 2020
G.f.: Sum_{n >= 1} q^(n^2)*( n^4 - (2*n^4 - 4*n^3 - 3*n^2 - n)*q^n - (8*n^3 - 4*n)*q^(2*n) + (2*n^4 + 4*n^3 - 3*n^2 + n)*q^(3*n) - n^4*q^(4*n) )/(1 - q^n)^4. Apply the operator x*d/dx twice, followed by the operator q*d/dq once, to equation 5 in Arndt and then set x = 1. - Peter Bala, Jan 21 2021
a(n) = Sum_{k = 1..n} sigma_3( gcd(k, n) ) = Sum_{d divides n} sigma_3(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k <= n} sigma_1( gcd(i, j, k, n) ) = Sum_{d divides n} sigma_1(d) * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 22 2024

A034714 Dirichlet convolution of squares with themselves.

Original entry on oeis.org

1, 8, 18, 48, 50, 144, 98, 256, 243, 400, 242, 864, 338, 784, 900, 1280, 578, 1944, 722, 2400, 1764, 1936, 1058, 4608, 1875, 2704, 2916, 4704, 1682, 7200, 1922, 6144, 4356, 4624, 4900, 11664, 2738, 5776, 6084, 12800, 3362, 14112, 3698, 11616, 12150, 8464

Offset: 1

Erich Friedman

Bruno Berselli, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000005, A000290, A001620, A038040, A134576, A319085 (partial sums).

[n^2*NumberOfDivisors(n): n in [1..50]]; // Bruno Berselli, Feb 12 2014

A034714 := proc(n) n^2*numtheory[tau](n) ; end proc:
seq(A034714(n),n=1..20) ; # R. J. Mathar, Feb 03 2011

A034714[n_]:=DivisorSigma[0,n]*n^2; Array[A034714, 50] (* Enrique Pérez Herrero, Jun 26 2011 *)

A034714(n)=numdiv(n)*n^2 \\ Enrique Pérez Herrero, Jun 26 2011

Dirichlet g.f.: zeta^2(s-2).
Equals n^2*tau(n), where tau(n) = A000005(n) = number of divisors of n. - Jon Perry, Aug 28 2005
Multiplicative with a(p^e) = (e+1)p^(2e). - Mitch Harris, Jun 27 2005
Row sums of triangle A134576. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} k^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = n * A038040(n). - Torlach Rush, Feb 01 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (-p^2 * log(1 - 1/p^2)) = 1.27728092754165872535305748273941301416624226497497308879403022758421224... - Vaclav Kotesovec, Sep 19 2020
G.f.: Sum_{n >= 1} q^(n^2)*( n^4*q^(3*n) - n^2*(n^2 + 4*n - 2)*q^(2*n) - n^2*(n^2 - 4*n - 2)*q^n + n^4 )/(1 - q^n)^3 - apply the operator q*d/dq twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
Sum_{k=1..n} a(k) ~ (n^3/3) * (log(n) + 2*gamma - 1/3), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 02 2023
a(n) = Sum_{1 <= i, j <= n} sigma_2( gcd(i, j, n) ) = Sum_{d divides n} sigma_2(d) * J_2(n/d), where sigma_2(n) = A001157(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024

A034761 Dirichlet convolution of sigma(n) with itself.

Original entry on oeis.org

1, 6, 8, 23, 12, 48, 16, 72, 42, 72, 24, 184, 28, 96, 96, 201, 36, 252, 40, 276, 128, 144, 48, 576, 98, 168, 184, 368, 60, 576, 64, 522, 192, 216, 192, 966, 76, 240, 224, 864, 84, 768, 88, 552, 504, 288, 96, 1608, 178, 588, 288, 644, 108, 1104, 288, 1152, 320, 360

Offset: 1

Erich Friedman

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A000005, A000203 (sigma), A001620, A073002, A074962, A134577.

f[p_, e_] := ((e + 1)*p^(e + 3) - (e + 3)*(p^(e + 2) - p + 1) + 2)/(p - 1)^3; f[2, e_] := (e - 1)*2^(e + 2) + e + 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 16 2022 *)

Dirichlet g.f.: zeta^2(s)*zeta^2(s-1).
Multiplicative with a(2^e) = (e-1) 2^(e+2) + e + 5, a(p^e) = ((1+e)p^(e+3) - (3+e)(p^(e+2)-p+1) + 2)/(p-1)^3, p > 2. - Mitch Harris, Jun 27 2005 [corrected by Amiram Eldar, Oct 16 2022 and Sep 12 2023]
Equals A134577 * A000005. - Gary W. Adamson, Nov 02 2007
Also the Dirichlet convolution A000005 by A038040. - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (2*Pi^2 * log(n) + (4*gamma - 1)*Pi^2 + 24*zeta'(2)) / 144, where gamma is the Euler-Mascheroni constant A001620 and Zeta'(2) = A073002. Equivalently, Sum_{k=1..n} a(k) ~ Pi^4 * n^2 * (2*log(n) - 1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) / 144, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 28 2019

A324121 a(n) = gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 2, 1, 1, 2, 2, 4, 2, 8, 12, 1, 2, 3, 2, 6, 4, 4, 2, 12, 1, 2, 4, 56, 2, 24, 2, 3, 12, 2, 4, 1, 2, 4, 4, 10, 2, 48, 2, 12, 6, 8, 2, 4, 3, 3, 12, 2, 2, 24, 4, 8, 4, 2, 2, 24, 2, 8, 2, 1, 4, 48, 2, 6, 12, 16, 2, 3, 2, 2, 2, 4, 4, 24, 2, 2, 1, 2, 2, 112, 4, 4, 12, 4, 2, 18, 28, 24, 4, 8, 20, 36, 2, 3, 6, 1, 2, 24, 2, 2, 24

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Records 1, 2, 12, 56, 112, 120, 336, 720, 992, 2016, 4368, 8640, 14880, 16256, 26208, 59520, 78624, 120960, 131040, 191520, 227584, 297600, ... occur at positions: 1, 3, 6, 28, 84, 120, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 27846, 30240, 32760, 55860, 105664, 117800, ... . Note that A001599 is not a subsequence of the latter, as at least 18620 (present in A001599) is missing.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..117800

Cf. A000005, A000203, A001599, A038040, A106315, A106316, A156552, A324045, A324046, A324047, A324058, A324122.

Table[GCD[n DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Feb 17 2023 *)

PARI
```
A324121(n) = gcd(sigma(n),n*numdiv(n));
```

a(n) = gcd(A000203(n), A038040(n)).

a(n) = A324058(A156552(n)).

A327166 Number of divisors d of n for which A000005(d)*d is equal to n, where A000005(x) gives the number of divisors of x.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Antti Karttunen, Sep 19 2019

nonn

a(n) tells how many times in total n occurs in A038040.

			108 has the following twelve divisors: [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108]. Of these, only d=18 and d=27 are such that d*A000005(d) = 108, as 18*6 = 27*4 = 108. Thus a(108) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A038040.

Cf. also A327153, A327169.

Table[Sum[If[d*DivisorSigma[0, d] == n, 1, 0], {d, Divisors[n]}], {n, 1, 120}] (* Vaclav Kotesovec, Jul 23 2022 *)

A327166(n) = sumdiv(n,d,(d*numdiv(d))==n);

a(n) = Sum_{d|n} [A000005(d)*d == n], where [ ] is the Iverson bracket.

A053650 Cototient function of n^2.

Original entry on oeis.org

0, 2, 3, 8, 5, 24, 7, 32, 27, 60, 11, 96, 13, 112, 105, 128, 17, 216, 19, 240, 189, 264, 23, 384, 125, 364, 243, 448, 29, 660, 31, 512, 429, 612, 385, 864, 37, 760, 585, 960, 41, 1260, 43, 1056, 945, 1104, 47, 1536, 343, 1500, 969, 1456, 53, 1944, 825, 1792, 1197

Offset: 1

Labos Elemer, Feb 18 2000

Seems to be invertible like n*Phi(n). Compare with A002618, A038040.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A038040.

Cf. A001248, A002618, A053650, A053192, A053193, A036689, A229099.

List([1..60], n-> n*(n- Phi(n)) ); # G. C. Greubel, May 18 2019

a053650 = a051953 . a000290  -- Reinhard Zumkeller, Jan 21 2014

[n*(n-EulerPhi(n)): n in [1..60]]; // Vincenzo Librandi, Jul 27 2016

Table[n(n-EulerPhi[n]), {n, 60}] (* Michael De Vlieger, Jul 26 2016 *)

a(n) = n^2 - eulerphi(n^2) \\ Michel Marcus, Jul 27 2013

[n*(n - euler_phi(n)) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = n*(n - phi(n)) = n^2 - n*phi(n) = Cototient(n^2) = A051953(A000290(n)).
a(n) = n^2 - A002618(n).
For p prime, Cototient(p)=1 and a(p)=p.
a(n) = n*cototient(n) = n*A051953(n). - Omar E. Pol, Nov 22 2012
Dirichlet g.f.: zeta(s-2)*(1 - 1/zeta(s-1)). - Ilya Gutkovskiy, Jul 26 2016
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023

A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

Original entry on oeis.org

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55

Offset: 1

Alonso del Arte, May 26 2011

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A191161, Sequence Machine.

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).

Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)

a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011
From Antti Karttunen, Nov 22 2024: (Start)
Following formulas were conjectured by Sequence Machine:
For n > 1, a(n) = A191150(n) + A074206(n).
a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.
a(n) = Sum_{d|n} A330575(d).
a(n) = Sum_{d|n} d*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A074206(n/d).
a(n) = Sum_{d|n} A007429(d)*A174725(n/d).
a(n) = Sum_{d|n} A000010(d)*A253249(n/d).
a(n) = Sum_{d|n} A038040(d)*A323912(n/d).
a(n) = Sum_{d|n} A060640(d)*A323910(n/d).
(End)

A245214 Numbers k such that A245212(k) < 0.

Original entry on oeis.org

144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 540, 576, 600, 648, 672, 720, 768, 792, 840, 864, 900, 936, 960, 1008, 1056, 1080, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1536, 1560, 1584, 1620, 1632, 1680, 1728, 1800, 1824, 1848, 1872, 1920, 1944, 1980, 2016, 2040, 2100, 2112, 2160, 2240

Offset: 1

Jaroslav Krizek, Jul 23 2014

nonn

If d are divisors of k then values of sequence A245212(k) are by bending moments in point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length k with bracket in point 0 by the schema: A245212(k) = (k * tau(k)) - Sum_{(d

Numbers k such that A038040(k) = k * tau(k) < A245211(k) = Sum_{(d

From Amiram Eldar, Jul 19 2024: (Start)

Numbers whose divisors have a mean abundancy index that is larger than 2.

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 24, 243, 2571, 25583, 254794, 2551559, 25514104, 255112225, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02551... .

The least odd term in this sequence is a(276918705) = 10854718875. (End)

			Number 144 is in sequence because 144 * tau(144) = 2160  < Sum_{(d<144) | 144} (d * tau(d)) = 2226.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A038040, A245211, A245212, A245213.

[n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt 0]

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2500], s[#] > 2 &]  (* Amiram Eldar, Jul 19 2024 *)

isok(n) = (n*numdiv(n) - sumdiv(n, d, (dMichel Marcus, Aug 06 2014

is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2;} \\ Amiram Eldar, Jul 19 2024

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cp's OEIS Frontend

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(i*j).

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A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(i*j).

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A328259 a(n) = n * sigma_2(n).

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A034714 Dirichlet convolution of squares with themselves.

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A034761 Dirichlet convolution of sigma(n) with itself.

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A324121 a(n) = gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

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A327166 Number of divisors d of n for which A000005(d)*d is equal to n, where A000005(x) gives the number of divisors of x.

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A053650 Cototient function of n^2.

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A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).