A294361 -id:A294361 - 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

A294362 E.g.f.: exp(Sum_{n>=1} sigma_2(n) * x^n).

Original entry on oeis.org

1, 1, 11, 91, 1105, 13841, 230731, 3955771, 80483201, 1738065025, 41800101931, 1070731623611, 29804263624081, 878224530964561, 27672361220570795, 919409968480087771, 32304618825218432641, 1191168445737728717441, 46119903359374012564171

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 14 2017: (Start)
It appears that the sequence taken modulo 10 is periodic with period (1, 1, 1, 1, 5) of length 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+4) = 0 (mod 5); a(7*n+3) == 0 (mod 7); a(11*n+2) == 0 (mod 11); a(13*n+3) == 0 (mod 13); a(17*n+4) == 0 (mod 17); a(19*n+12) == 0 (mod 19). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..417
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), A294361 (k=1), this sequence (k=2).

Cf. A001157.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[2, 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, 2)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001157(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(k^2*x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
a(n) ~ (3*Zeta(3))^(1/8) * exp(2^(9/4) * Zeta(3)^(1/4) * n^(3/4) / 3^(3/4) - n^(1/4) / (2^(9/4) * 3^(5/4) * Zeta(3)^(1/4)) - n) * n^(n - 1/8) / 2^(7/8). - Vaclav Kotesovec, Sep 04 2018

A294403 E.g.f.: exp(-Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, -1, -5, -7, 1, 839, 4171, 54305, 102817, -4303441, -74521349, -1595325271, -20768141855, -222701825737, 1485790534411, 65580347824529, 2880129557707201, 67631429234674655, 1543424936566399867, 23542870556917468889, 119940955037901088321

Offset: 0

Seiichi Manyama, Oct 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..439

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): A294402 (k=0), this sequence (k=1), A294404 (k=2).

Cf. A000203, A010815, A069097, A294361.

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

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000203(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)^2. - 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.

A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A061255, A061256, A006171.
Cf. A299069, A192065, A107742.
Cf. A294361, A294363.

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

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

A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

Original entry on oeis.org

1, 6, 1, 24, 18, 1, 168, 204, 36, 1, 720, 2280, 780, 60, 1, 8640, 25200, 14400, 2100, 90, 1, 40320, 292320, 252000, 58800, 4620, 126, 1, 604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1, 4717440, 46811520, 76265280, 35743680, 6335280, 474768, 15624, 216, 1

Offset: 1

Seiichi Manyama, Nov 13 2020

			exp(Sum_{n>0} u*sigma(n)*x^n) = 1 + u*x + (6*u+u^2)*x^2/2! + (24*u+18*u^2+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       6,       1;
      24,      18,       1;
     168,     204,      36,       1;
     720,    2280,     780,      60,      1;
    8640,   25200,   14400,    2100,     90,    1;
   40320,  292320,  252000,   58800,   4620,  126,   1;
  604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1;
  ...

Peter Luschny, The Bell transform.

Column k=1..2 give n! * sigma(n), (n!/2) * A000385(n-1).
Rows sum give A294361.
Cf. A000203 (sigma(n)), A008298, A338864, A338871.

T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * j! * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, exp(j*x^j/(1-x^j+x*O(x^n)))^u), n), k)}

a(n) = if(n<1, 0, n!*sigma(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

E.g.f.: exp(Sum_{n>0} u*sigma(n)*x^n).
T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} k*sigma(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n,k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j).

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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A294362 E.g.f.: exp(Sum_{n>=1} sigma_2(n) * x^n).

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A294403 E.g.f.: exp(-Sum_{n>=1} sigma(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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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

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

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A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

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A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.