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 1-5 of 5 results.

A100279 a(n) = A100107(A000032(n)).

Original entry on oeis.org

4, 1, 5, 11, 30, 200, 5880, 1149852, 6643838880, 7639424866275970, 50755107359004694925071660, 387739824812222466915538827541705412334750, 19679776435706023589554719270187917683310683639911856695096924149852
Offset: 0

Views

Author

Jonathan Vos Post, Dec 26 2004

Keywords

Crossrefs

Cf. A000032, A100107.

Programs

Extensions

Corrected and extended by R. J. Mathar, Aug 21 2007
Name corrected using formula by R. J. Mathar, Andrey Zabolotskiy, Apr 30 2021

A130878 Inverse Moebius transform of A100107.

Original entry on oeis.org

1, 5, 6, 16, 13, 36, 31, 74, 87, 155, 201, 402, 523, 911, 1398, 2339, 3573, 5997, 9351, 15438, 24546, 40011, 64081, 104544, 167786, 272495, 439372, 712452, 1149853, 1863588, 3010351, 4875451, 7881606, 12759195, 20633323, 33397854, 54018523, 87422511, 141423378
Offset: 1

Views

Author

R. J. Mathar, Aug 21 2007

Keywords

Comments

Or, the inverse Moebius transform of the inverse Moebius transform of the Lucas numbers A000032.

Crossrefs

Cf. A000032, A100107, A100279.

Programs

  • Maple
    A000032 := proc(n) option remember; if n =0 then 2; elif n = 1 then 1; else A000032(n-1)+A000032(n-2) ; fi ; end: A100107 := proc(n) option remember ; local a,dvs,d ; a := 0: dvs := numtheory[divisors](n) ; for d in dvs do a := a+ A000032(d) ; od: RETURN(a) ; end: a := proc(n) local a,dvs,d ; a := 0: dvs := numtheory[divisors](n) ; for d in dvs do a := a+ A100107(d) ; od: RETURN(a) ; end: seq(a(n),n=1..100);
# second Maple program:
a:= ((p-> j-> add(p(d), d=numtheory[divisors](j)))@@2)
     (n-> (<<1|1>, <1|0>>^n.<<2, -1>>)[1, 1]):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 23 2024
  • Mathematica
    A100107[n_] := LucasL /@ Divisors[n] // Total;
a[n_] := A100107 /@ Divisors[n] // Total;
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 23 2024 *)

Formula

a(n) = Sum_{d|n} A100107(d).

A007435 Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...

Original entry on oeis.org

1, 2, 3, 5, 6, 12, 14, 26, 37, 62, 90, 159, 234, 392, 618, 1013, 1598, 2630, 4182, 6830, 10962, 17802, 28658, 46548, 75031, 121628, 196455, 318206, 514230, 832722, 1346270, 2179322, 3524670, 5704486, 9227484, 14933129, 24157818, 39092352, 63246222, 102341006
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

For p prime, a(p) == k (mod p) where k = 0 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 1 if p = 5. - Michael Somos, Apr 15 2012

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 12*x^6 + 14*x^7 + 26*x^8 + 37*x^9 + 62*x^10 + ...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000045, A051731, A127647, A245282, A108046, A034772, A100107, A108031, A008683.

Programs

  • Maple
    with(numtheory): seq(add(combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 12 2025
  • Mathematica
    Table[Plus @@ Map[Function[d, Fibonacci[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
a[n_] := DivisorSum[n, Fibonacci]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* Michael Somos, Apr 15 2012 */

Formula

Row sums of A051731 * A127647. - Gary W. Adamson, Jan 22 2007
G.f.: Sum_{k>0} Fibonacci(k)*x^k/(1-x^k) = Sum_{k>0} x^k/(1-x^k-x^(2*k)). - Vladeta Jovovic, Dec 17 2002
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(Fibonacci(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 20 2018
a(n) ~ 5^(-1/2) * phi^n, where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2018
From Ridouane Oudra, Apr 12 2025 : (Start)
a(n) = Sum_{d|n} Fibonacci(d).
a(n) = Sum_{d|n} mu(d)*A034772(n/d).
a(n) = A245282(n) - A108046(n).
a(n) = 2*A245282(n) - A100107(n).
a(n) = (A108031(n) + A108046(n))/2. (End)

Extensions

More terms from Joerg Arndt, Aug 14 2012

A108031 Inverse Moebius transform of Lucas numbers (A000032).

Original entry on oeis.org

2, 3, 5, 7, 9, 17, 20, 36, 52, 86, 125, 220, 324, 542, 855, 1400, 2209, 3635, 5780, 9439, 15150, 24602, 39605, 64328, 103691, 168086, 271495, 439750, 710649, 1150794, 1860500, 3011749, 4870975, 7883406, 12752070, 20637077, 33385284, 54024302
Offset: 1

Views

Author

Emeric Deutsch, May 31 2005

Keywords

Examples

			a(4)=7 because the divisors of 4 are 1,2,4 and the first, second and fourth Lucas numbers are 2, 1 and 4, respectively, having sum 7.

Crossrefs

Cf. A000032, A001622, A100107.

Programs

  • Maple
    with(combinat): with(numtheory): f:=n->2*fibonacci(n)-fibonacci(n-1): g:=proc(n) local div: div:=divisors(n): sum(f(div[j]),j=1..tau(n)) end: seq(g(n),n=1..45);
  • Mathematica
    Table[Total[LucasL[#]&/@(Divisors[n]-1)],{n,40}] (* Harvey P. Dale, Dec 08 2014 *)

Formula

a(n) ~ phi^(n-1), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 10 2021

A309729 Expansion of Sum_{k>=1} x^k/(1 - x^k - 2*x^(2*k)).

Original entry on oeis.org

1, 2, 4, 7, 12, 26, 44, 92, 175, 354, 684, 1396, 2732, 5506, 10938, 21937, 43692, 87578, 174764, 349884, 699098, 1398786, 2796204, 5593886, 11184823, 22372354, 44739418, 89483996, 178956972, 357925242, 715827884, 1431677702, 2863312218, 5726666754, 11453246178, 22906581193
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

Comments

Inverse Moebius transform of Jacobsthal numbers (A001045).

Crossrefs

Cf. A001045, A007435, A055895, A100107, A104723, A256281.

Programs

  • Maple
    seq(add(2^d-(-1)^d, d=numtheory:-divisors(n))/3, n=1..50); # Robert Israel, Aug 14 2019
  • Mathematica
    nmax = 36; CoefficientList[Series[Sum[x^k/(1 - x^k - 2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[(1/3) Sum[(2^d - (-1)^d), {d, Divisors[n]}], {n, 1, 36}]
  • PARI
    a(n)={sumdiv(n, d, 2^d - (-1)^d)/3} \\ Andrew Howroyd, Aug 14 2019
  • Python
    n = 1
while n <= 36:
    s, d = 0, 1
    while d <= n:
        if n%d == 0:
            s = s+2**d-(-1)**d
        d = d+1
    print(n,s//3)
n = n+1 # A.H.M. Smeets, Aug 14 2019

Formula

G.f.: Sum_{k>=1} A001045(k) * x^k/(1 - x^k).
a(n) = (1/3) * Sum_{d|n} (2^d - (-1)^d).
Showing 1-5 of 5 results.

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

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