A294362 -id:A294362 - cp's OEIS frontend

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 13 2017: (Start)
The terms of the sequence appear to be of the form 4*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)
a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 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

A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, 1, 7, 43, 409, 3841, 50431, 648187, 10347793, 170363809, 3200390551, 62855417131, 1371594161257, 31147757782753, 768384638386639, 19814802390611131, 545309251861956001, 15661899520801953217, 475833949719419469223, 15042718034104688144299

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 14 2017: (Start)
The terms of the sequence appear to be of the form 6*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), this sequence (k=1), A294362 (k=2).

Cf. A000203, A053529, A274804, A061256, A192065.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 04 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3)/2 - 3^(1/3) * n^(1/3) / (2*Pi^(2/3)) + 1/24 - 1/(8*Pi^2) - n) * n^(n - 1/6) / 3^(2/3). - Vaclav Kotesovec, Sep 04 2018

A294404 E.g.f.: exp(-Sum_{n>=1} sigma_2(n) * x^n).

Original entry on oeis.org

1, -1, -9, -31, -23, 3399, 41311, 473129, 1284081, -79051537, -2447228249, -52444297071, -712806368999, -2221410364681, 331443685309647, 15068893004257049, 460836352976093281, 10298306504802529119, 122928784866003823831, -3359583359629857247807

Offset: 0

Seiichi Manyama, Oct 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..426

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): A294402 (k=0), A294403 (k=1), this sequence (k=2).

Cf. A001157, A073592, A294362, A343497.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sigma(k, 2)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A001157(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j)^3. - Ilya Gutkovskiy, Aug 17 2021

A294296 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} sigma_k(j) * x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 7, 25, 1, 1, 11, 43, 193, 1, 1, 19, 91, 409, 1481, 1, 1, 35, 223, 1105, 3841, 16021, 1, 1, 67, 595, 3505, 13841, 50431, 167665, 1, 1, 131, 1663, 12193, 60841, 230731, 648187, 2220065, 1, 1, 259, 4771, 44689, 297761, 1340851, 3955771

Offset: 0

Seiichi Manyama, Oct 30 2017

			Square array A(n,k) begins:
      1,    1,     1,     1,      1, ...
      1,    1,     1,     1,      1, ...
      5,    7,    11,    19,     35, ...
     25,   43,    91,   223,    595, ...
    193,  409,  1105,  3505,  12193, ...
   1481, 3841, 13841, 60841, 297761, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A294363, A294361, A294362.
Rows n=0-1 give A000012.
Main diagonal gives A294388.
Cf. A144048.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*sigma_k(j)*A(n-j,k)/(n-j)! for n > 0.

A352694 Expansion of e.g.f. exp(Sum_{k>=1} sigma_2(k) * x^k/k!).

Original entry on oeis.org

1, 1, 6, 26, 167, 1157, 9372, 82742, 806872, 8487255, 96086764, 1159845766, 14866684968, 201266031865, 2867695938970, 42849364911878, 669517721182731, 10910196881874549, 184997231064875867, 3257297876661453487, 59443905364431491367, 1122496527274459462803

Offset: 0

Seiichi Manyama, Mar 29 2022

nonn

Exponential transform of A001157.

Seiichi Manyama, Table of n, a(n) for n = 0..505

Cf. A295739, A274804.

Cf. A001157, A294362, A352693.

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sigma(k, 2)*x^k/k!))))

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, 2)*binomial(n-1, k-1)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} sigma_2(k) * binomial(n-1,k-1) * a(n-k).

A352842 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k).

Original entry on oeis.org

1, 1, 11, 199, 7585, 427961, 37901851, 4526311231, 729098029409, 149311985624785, 38243144308952971, 11913301283967428951, 4445712423354285230401, 1954806416110914007773769, 1000799932457357582959443035, 589931632494798210345741193231

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A294363, A294361, A294362.

nmax = 20; CoefficientList[Series[E^(Sum[DivisorSigma[k, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sigma(k, k)*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sigma(k, k)*a(n-k)/(n-k)!));

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * sigma_k(k) * a(n-k)/(n-k)!.

a(n) ~ n! * n^n. - Vaclav Kotesovec, Apr 15 2022

cp's OEIS Frontend

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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

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A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

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A294404 E.g.f.: exp(-Sum_{n>=1} sigma_2(n) * x^n).

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A294296 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} sigma_k(j) * x^j).

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A352694 Expansion of e.g.f. exp(Sum_{k>=1} sigma_2(k) * x^k/k!).

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A352842 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k).

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