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 1-6 of 6 results.

A112962 Sum(mu(i)*phi(j): i+j=n), with mu=A008683 and phi=A000010.

Original entry on oeis.org

0, 1, 0, 0, -1, -1, -4, -2, -5, -8, -5, -8, -9, -11, -10, -24, 1, -21, -11, -23, -15, -37, 4, -42, -11, -38, -7, -49, 6, -63, -12, -44, -3, -81, 10, -106, 7, -49, -8, -92, 15, -103, 2, -72, -5, -114, 41, -140, -3, -114, 8, -113, 49, -179, 3, -135, 27, -131, 46, -218, -7, -99, 32, -185, 72, -259, 50, -104, 23, -211, 52, -248, 43, -153
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Examples

			a(5)=mu(1)*phi(4)+mu(2)*phi(3)+mu(3)*phi(2)+mu(4)*phi(1) = 1*2 - 1*2 - 1*1 + 0*1 = -1.

Crossrefs

Cf. A002088, A065093, A068341, A112963, A112964, A112966, A112968.

Programs

  • Maple
    with(numtheory); f:=n->add(phi(i)*mobius(n-i),i=1..n-1);
  • PARI
    a(n)=sum(i=1,n-1,moebius(i)*eulerphi(n-i)) \\ Charles R Greathouse IV, Feb 21 2013

Extensions

Corrected by N. J. A. Sloane, Mar 01 2006

A340995 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the Euler totient function phi; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 8, 9, 4, 1, 0, 2, 16, 19, 14, 5, 1, 0, 6, 20, 42, 36, 20, 6, 1, 0, 4, 36, 72, 89, 60, 27, 7, 1, 0, 6, 44, 134, 184, 165, 92, 35, 8, 1, 0, 4, 68, 210, 376, 391, 279, 133, 44, 9, 1, 0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1
Offset: 0

Views

Author

Alois P. Heinz, Feb 01 2021

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  4,  8,   9,   4,   1;
  0,  2, 16,  19,  14,   5,   1;
  0,  6, 20,  42,  36,  20,   6,   1;
  0,  4, 36,  72,  89,  60,  27,   7,   1;
  0,  6, 44, 134, 184, 165,  92,  35,   8,  1;
  0,  4, 68, 210, 376, 391, 279, 133,  44,  9,  1;
  0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1;
  ...

Links

Crossrefs

Columns k=0-2 give (offsets may differ): A000007, A000010, A065093.
Row sums give A159929.
T(2n,n) gives A340994.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[phi](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, EulerPhi[n]], With[{q = Quotient[k, 2]},
     Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)

Formula

T(n,k) = [x^n] (Sum_{j>=1} phi(j)*x^j)^k.

A307308 Self-composition of the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 6, 15, 42, 106, 280, 702, 1778, 4398, 10910, 26678, 65172, 157656, 380524, 912846, 2185906, 5216588, 12433166, 29564544, 70189672, 166245574, 392909240, 926290066, 2178881218, 5114469170, 11985221654, 28049398284, 65588182636, 153277006212, 358073997608
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2019

Keywords

Links

Crossrefs

Cf. A000010, A008683, A010554, A065093, A307309.

Programs

  • Mathematica
    g[x_] := g[x] = Sum[MoebiusMu[k] x^k/(1 - x^k)^2, {k, 1, 31}]; a[n_] := a[n] = SeriesCoefficient[g[g[x]], {x, 0, n}]; Table[a[n], {n, 31}]

Formula

G.f.: g(g(x)), where g(x) = Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2 is the g.f. of A000010.

A113793 Triangle read by rows: T(n,m) = phi(n - m + 1) * phi(m), n >= 1, m >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 4, 2, 6, 2, 8, 4, 8, 2, 6, 4, 6, 4, 8, 8, 4, 6, 4, 6, 4, 12, 4, 16, 4, 12, 4, 6, 4, 6, 8, 12, 8, 8, 12, 8, 6, 4, 10, 4, 12, 8, 24, 4, 24, 8, 12, 4, 10, 4, 10, 8, 12, 16, 12, 12, 16, 12, 8, 10, 4, 12, 4, 20, 8, 24, 8, 36, 8, 24, 8, 20, 4, 12
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 25 2008

Keywords

Examples

			{1},
{1, 1},
{2, 1, 2},
{2, 2, 2, 2},
{4, 2, 4, 2, 4},
{2, 4, 4, 4, 4, 2},
{6, 2, 8, 4, 8, 2, 6},
{4, 6, 4, 8, 8, 4, 6, 4},
{6, 4, 12, 4, 16, 4, 12, 4, 6},
{4, 6, 8, 12, 8, 8, 12, 8, 6, 4},
{10, 4, 12, 8, 24, 4, 24, 8, 12, 4, 10}

Crossrefs

Column 1 and leading diagonal give A000010.
Middle diagonal gives A127473.
Row sums give A065093.

Programs

  • Mathematica
    T[n_, m_] = EulerPhi[n - m + 1]*EulerPhi[m + 1]; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

T(n,m) = A000010(m)*A000010(n-m+1), n >= 1, m >= 1. - Omar E. Pol, Jan 14 2025

Extensions

Name corrected and more terms added by Omar E. Pol, Jan 14 2025

A307502 Self-convolution of the Dedekind psi function (A001615).

Original entry on oeis.org

0, 1, 6, 17, 36, 64, 108, 172, 240, 340, 444, 612, 744, 980, 1164, 1504, 1704, 2172, 2388, 2964, 3288, 3968, 4272, 5272, 5520, 6624, 7104, 8276, 8640, 10404, 10572, 12480, 13032, 14988, 15300, 18204, 18048, 21004, 21636, 24616, 24648, 29036, 28452, 32768, 33552, 37488
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 11 2019

Keywords

Links

Crossrefs

Cf. A001615, A008683, A065093, A307309.

Programs

  • Mathematica
    Rest[nmax = 46; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k)^2, {k, 1, nmax}]^2, {x, 0, nmax}], x]]
psi[n_] := psi[n] = Sum[MoebiusMu[n/d]^2 d, {d, Divisors @ n}]; a[n_] := a[n] = Sum[psi[k] psi[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 46}]

Formula

G.f.: (Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2)^2.
a(n) = Sum_{k=1..n-1} A001615(k)*A001615(n-k).
Conjecture: Sum_{k=1..n} a(k) ~ 75 * n^4 / (8 * Pi^4). - Vaclav Kotesovec, Aug 20 2025

A330148 a(n) = Sum_{k=1..n} binomial(n,k) * phi(k) * phi(n - k + 1), where phi = A000010.

Original entry on oeis.org

1, 3, 11, 30, 94, 238, 692, 1596, 4536, 9350, 27840, 52884, 149668, 294838, 782432, 1463224, 4095792, 7460274, 20229356, 36847380, 100317284, 170262974, 492659240, 814679680, 2184447760, 3965791284, 9988168320, 17883230712, 49362800340, 80674575956, 213420581248
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 03 2019

Keywords

Crossrefs

Cf. A000010, A007318, A065093, A306988.

Programs

  • Magma
    [&+[Binomial(n,k)*EulerPhi(k)*EulerPhi(n-k+1):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Dec 03 2019
  • Mathematica
    Table[Sum[Binomial[n, k] EulerPhi[k] EulerPhi[n - k + 1], {k, 1, n}], {n, 1, 31}]
nmax = 31; CoefficientList[Series[(1/2) D[Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = sum(k=1, n, binomial(n,k)*eulerphi(k)*eulerphi(n-k+1)); \\ Michel Marcus, Dec 03 2019

Formula

E.g.f.: (1/2) * d/dx (Sum_{k>=1} phi(k) * x^k / k!)^2.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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