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.

Previous Showing 11-15 of 15 results.

A160619 a(n) = Sum_{d|n} phi(n/d)*2^(d+1), with a(0) = 0.

Original entry on oeis.org

0, 4, 12, 24, 48, 80, 168, 280, 576, 1080, 2160, 4136, 8448, 16432, 33096, 65760, 131712, 262208, 525672, 1048648, 2099520, 4194960, 8392824, 16777304, 33564096, 67109200, 134234256, 268437672, 536904480, 1073741936, 2147554080, 4294967416, 8590066944
Offset: 0

Views

Author

N. J. A. Sloane, Nov 21 2009

Keywords

Links

Crossrefs

Cf. A034738, A053635, A161219.

Programs

  • Mathematica
    a[n_]:= If[n<1, 0, Sum[EulerPhi[n/d] 2^(d+1), {d, Divisors[n]}]]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, May 06 2018 *)
  • PARI
    a(n) = if (n==0, 0, sumdiv(n, d, eulerphi(n/d)*2^(d+1))); \\ Michel Marcus, May 07 2018; corrected Jun 14 2022

Formula

a(n) = 4*A034738(n). - Michel Marcus, May 07 2018

Extensions

Name edited by Michel Marcus, Jun 14 2022

A160620 a(n) = Sum_{d|n} phi(n/d)^2*2^d.

Original entry on oeis.org

0, 2, 6, 16, 28, 64, 96, 200, 320, 616, 1152, 2248, 4304, 8480, 16728, 33152, 66048, 131584, 263160, 524936, 1050176, 2098240, 4196952, 8389576, 16782976, 33555744, 67117920, 134220712, 268453360, 536872480, 1073780352, 2147485448, 4295034880, 8589944384
Offset: 0

Views

Author

N. J. A. Sloane, Nov 21 2009

Keywords

Links

Crossrefs

Cf. A053635.
Cf. A000079, A127473.

Programs

  • Maple
    A160620 := proc(n)
    if n =0 then
        0;
    else
        add((numtheory[phi](n/d))^2*2^d,d=numtheory[divisors](n)) ;
    end if ;
end proc: # R. J. Mathar, Jun 24 2021
  • Mathematica
    a[n_]:= If[n<1, 0, Sum[EulerPhi[n/d]^2 * 2^d, {d, Divisors[n]}]]; Table[a[n], {n,0,50}] (* G. C. Greubel, May 06 2018 *)
  • PARI
    a(n) = if (n, sumdiv(n, d, eulerphi(n/d)^2*2^d), 0); \\ Michel Marcus, May 07 2018, Jun 22 2021

Formula

Dirichlet (convolution) product of A127473 and A000079. - R. J. Mathar, Jun 24 2021

A161219 a(n) = (1/n) * Sum_{d|n} phi(n/d)*2^(d+1).

Original entry on oeis.org

4, 6, 8, 12, 16, 28, 40, 72, 120, 216, 376, 704, 1264, 2364, 4384, 8232, 15424, 29204, 55192, 104976, 199760, 381492, 729448, 1398504, 2684368, 5162856, 9942136, 19175160, 37025584, 71585136, 138547336, 268439592, 520602352, 1010588256, 1963413664, 3817763800
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2009

Keywords

Links

Crossrefs

Cf. A000031, A053635, A160619, A161217.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*2^(d+1))/n; \\ Michel Marcus, Apr 15 2021

Formula

G.f.: -2 * Sum_{k>=1} phi(k) * log(1 - 2*x^k) / k. - Ilya Gutkovskiy, Apr 15 2021
a(n) = 2*A000031(n). - Hugo Pfoertner, Jun 24 2021

A327030 a(n) = Sum_{d|n} phi(d)*(n/d)! for n > 0, a(0) = 0.

Original entry on oeis.org

0, 1, 3, 8, 28, 124, 732, 5046, 40352, 362898, 3628932, 39916810, 479002388, 6227020812, 87178296258, 1307674368272, 20922789928384, 355687428096016, 6402373706092350, 121645100408832018, 2432902008180269152, 51090942171709450128, 1124000727777647596830
Offset: 0

Views

Author

Peter Luschny, Aug 27 2019

Keywords

Comments

Dirichlet convolution of phi(n) and n! (n >= 1). - Richard L. Ollerton, May 09 2021

Links

Crossrefs

Cf. A000010, A000142, A027750.
Similar: A078392 (numbpart), A258171 (bell), A053635 (numbcomb), A181847 and A034738 (numbcomp), this sequence (numbperm).

Programs

  • Magma
    [0] cat [&+[EulerPhi(d)*Factorial(n div d):d in Divisors(n)]:n in [1..22]]; // Marius A. Burtea, Nov 13 2019
  • Magma
    [0] cat [&+[Factorial(Gcd(n,i)):i in [1..n]]:n in [1..22]]; // Marius A. Burtea, Nov 13 2019
  • Maple
    with(numtheory); A327030 := n -> add(phi(d)*(n/d)!, d = divisors(n)):
seq(A327030(n), n=0..22);
  • Mathematica
    a[0] = 0; a[n_] := DivisorSum[n, EulerPhi[#] * (n/#)! &]; Array[a, 23, 0] (* Amiram Eldar, May 24 2021 *)
  • PARI
    a(n) = if (n>0, sumdiv(n, d, eulerphi(d)*(n/d)!), 0); \\ Michel Marcus, Aug 28 2019

Formula

a(n) = Sum_{i=1..n} gcd(n,i)!. - Ridouane Oudra, Nov 13 2019

A346558 a(n) = Sum_{d|n} phi(n/d) * (2^d - 1).

Original entry on oeis.org

1, 4, 9, 20, 35, 78, 133, 280, 531, 1070, 2057, 4212, 8203, 16534, 32865, 65840, 131087, 262818, 524305, 1049740, 2097459, 4196390, 8388629, 16782024, 33554575, 67117102, 134218809, 268452212, 536870939, 1073777010, 2147483677, 4295033440, 8589938775, 17180000318, 34359739085
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 17 2021

Keywords

Crossrefs

Cf. A000010, A000031, A000225, A008965, A034738, A038199, A053635.

Programs

  • Mathematica
    Table[Sum[EulerPhi[n/d] (2^d - 1), {d, Divisors[n]}], {n, 1, 35}]
nmax = 35; CoefficientList[Series[Sum[EulerPhi[k] x^k/((1 - x^k) (1 - 2 x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(2^d - 1)); \\ Michel Marcus, Sep 17 2021

Formula

G.f.: Sum_{k>=1} phi(k) * x^k / ((1 - x^k) * (1 - 2*x^k)).
a(n) = Sum_{k=1..n} (2^gcd(n,k) - 1).
a(n) = n * (A000031(n) - 1) = n * A008965(n).
Dirichlet convolution of A000225 and A000010. - R. J. Mathar, Sep 30 2021
Previous Showing 11-15 of 15 results.

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

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