cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A308462 Expansion of e.g.f. exp(Sum_{k>=1} psi(k)*x^k/k), where psi() is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 1, 4, 18, 114, 810, 7560, 71820, 822780, 10086300, 139532400, 2035618200, 33149655000, 562448086200, 10416443637600, 202624824402000, 4207527414090000, 91475485119018000, 2114681171586984000, 50821588411117524000, 1289125346347418580000
Offset: 0

Views

Author

Ilya Gutkovskiy, May 28 2019

Keywords

Links

Crossrefs

Cf. A001615, A301511, A318917.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Product[1 + Boole[PrimeQ[d]]/d, {d, Divisors[k]}] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

Formula

log(a(n)/n!) ~ 2*sqrt(15*n)/Pi. - Vaclav Kotesovec, Oct 31 2024

A377507 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^2 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 12, 66, 690, 4860, 63000, 711900, 8876700, 131405400, 2160219600, 37553808600, 686750664600, 13805424032400, 278759396916000, 6445702905642000, 150985820419434000, 3825993309462324000, 99427990563910008000, 2724045313186016820000, 78032929885709378580000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 30 2024

Keywords

Links

Crossrefs

Cf. A318917, A377508, A377509.
Cf. A065464, A127473, A156302.

Programs

  • Mathematica
    nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^2*a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^2 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

log(a(n)/n!) ~ 3 * c^(1/3) * n^(2/3) / 2^(2/3), where c = Product_{p primes} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298...

A377508 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^3 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 20, 122, 2122, 15532, 284104, 3837500, 52963964, 1125315224, 20981180464, 500475045688, 10373180665720, 264908485440848, 6624880728277088, 185812008437953808, 5449866267968244496, 167510440639938875680, 5447433174773217714496, 177500241844579492474016
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 30 2024

Keywords

Links

Crossrefs

Cf. A318917, A377507, A377509.
Cf. A178933, A358714.

Programs

  • Mathematica
    nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^3*a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^3 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

log(a(n)/n!) ~ 2^(9/4) * c^(1/4) * n^(3/4) / 3^(3/4), where c = Product_{p primes} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.337187873791589971961692816152158244949154127758...

A377509 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^4 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 36, 234, 7290, 54540, 1408680, 23119740, 341788860, 11790437400, 231972879600, 8206299070200, 191673262380600, 6154270418696400, 206515993375692000, 6574758436640394000, 269828090984990538000, 9531096165082736244000, 411037724983993923816000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 30 2024

Keywords

Links

Crossrefs

Cf. A318917, A377507, A377508.
Cf. A205797, A361148.

Programs

  • Mathematica
    nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^4 * a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^4 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

log(a(n)/n!) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956...

A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 174, 278, 447, 707, 1122, 1766, 2729, 4213, 6482, 9880, 15069, 22799, 34290, 51378, 76777, 114365, 169324, 250162, 368505, 540575, 792042, 1154798, 1680385, 2439101, 3530308, 5103380, 7349875, 10564955, 15155752, 21696072, 31007949, 44199845
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Links

Crossrefs

Cf. A000010, A061255, A159929, A318917.
Cf. A279784, A306327, A316961.

Programs

  • Maple
    with(numtheory): a:=series(mul(1/(1-phi(k)*x^k),k=1..50),x=0,42): seq(coeff(a,x,n),n=0..41); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 41; CoefficientList[Series[Product[1/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d EulerPhi[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 18827.6460615531202942792897255332975807324818737172163... if mod(n,5) = 0
c = 18827.5079339024144115146595255453426552477117955925738... if mod(n,5) = 1
c = 18827.4967567108036710998657106724179082561779712900405... if mod(n,5) = 2
c = 18827.4818413568083742650057347700058389606441225811016... if mod(n,5) = 3
c = 18827.4547665561882994953942505862213438332903500716893... if mod(n,5) = 4
(End)

A328053 Expansion of e.g.f. log(1 + Sum_{k>=1} phi(k) * x^k / k!), where phi = Euler totient function (A000010).

Original entry on oeis.org

0, 1, 0, 1, -3, 8, -32, 166, -926, 5842, -42812, 348632, -3088388, 29871372, -314102574, 3554714938, -43057252520, 556487433400, -7644034688586, 111160926400032, -1706191876272876, 27567942738717360, -467712309003533398, 8312805777830133096
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

Comments

Logarithmic transform of A000010.

Links

Crossrefs

Cf. A000010, A300011, A318811, A318917.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 0, (b-> b(n)-add(a(j)
     *binomial(n, j)*j*b(n-j), j=1..n-1)/n)(numtheory[phi]))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 06 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = EulerPhi[n] - Sum[Binomial[n, k] EulerPhi[n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

Formula

a(n)/n! ~ -(-1)^n * d^n / n, where d = 0.8078801380543809482705136550646927880437760099284517780830096910529492372472... - Vaclav Kotesovec, Oct 17 2019

A353192 Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k) * x^k / k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 16, 110, 986, 10202, 126288, 1770120, 27939192, 489658632, 9455296896, 198951693360, 4537680805776, 111426422418768, 2931467216681856, 82273083792879744, 2453340521239749504, 77458777017799833216, 2581489882182061744128
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2022

Keywords

Crossrefs

Cf. A000010, A074930, A159929, A318917, A352887.

Programs

  • Mathematica
    phi[k_] := phi[k] = EulerPhi[k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * phi[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Apr 30 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*eulerphi(j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A074930(k) * binomial(n,k) * a(n-k).
Showing 1-7 of 7 results.

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