A181982 -id:A181982 - cp's OEIS frontend

A382488 The number of unitary 3-smooth divisors of n.

1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2

Offset: 1

Amiram Eldar, Mar 29 2025

Period 6: repeat [1, 2, 2, 2, 1, 4].
Decimal expansion of 407380/333333.
Continued fraction expansion of 10/(6 + sqrt(66)) (with offset 0).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A005117, A003586, A034444, A065331, A072078, A181982, A382487.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), this sequence (k = 2), A382489 (k = 3).

Table[{1, 2, 2, 2, 1, 4}, {12}] // Flatten

a(n) = [1, 2, 2, 2, 1, 4][(n-1) % 6 + 1];

Multiplicative with a(p^e) = 2 if p <= 3, and 1 otherwise.
a(n) = A034444(A065331(n)).
a(n) = A034444(n) if and only if n is 3-smooth (A003586).
a(n) = A072078(n) if and only if n is squarefree (A005117).
a(n) = abs(A181982(n+9)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.
G.f.: -(4*x^6 + x^5 + 2*x^4 + 2*x^3 +2*x^2 + x)/(x^6 - 1).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * zeta(s).

A236773 a(n) = n + floor( n^2/2 + n^3/3 ).

Original entry on oeis.org

0, 1, 6, 16, 33, 59, 96, 145, 210, 292, 393, 515, 660, 829, 1026, 1252, 1509, 1799, 2124, 2485, 2886, 3328, 3813, 4343, 4920, 5545, 6222, 6952, 7737, 8579, 9480, 10441, 11466, 12556, 13713, 14939, 16236, 17605, 19050, 20572, 22173, 23855, 25620, 27469

Offset: 0

Bruno Berselli, Feb 07 2014

This sequence follows A074148 and A042965, A236771.
The prime terms are 59, 829, 14939, 35759, 93719, 132409, 155219, 290399, 414179, 487463, ... .
If a(k) is prime then k == 1, 5, 7 or 11 (mod 12).
Third differences: 1, 2, 2, 2, 1, 4 repeated (unsigned terms of A181982).
Fourth differences: 1, 0, 0, -1, 3, -3 repeated (see A131193).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,1,-3,3,-1).

Cf. A074148: n+floor(n^2/2).
Cf. A042965: n+floor(1/2+n/3); A236771: n+floor(n/2+n^2/3).
Cf. A236772: floor(sum(i=1..n, n^i/i)).

Magma
```
[n+Floor(n^2/2+n^3/3): n in [0..50]];
```

I:=[0,1,6,16,33,59,96,145,210]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-6)-3*Self(n-7)+3*Self(n-8)-Self(n-9): n in [1..50]]; // Vincenzo Librandi, Feb 08 2014

seq(n+floor(n^2/2+n^3/3),n=0..43); # Paolo P. Lava, Aug 24 2018

Table[n + Floor[n^2/2 + n^3/3], {n, 0, 50}]
CoefficientList[Series[x (1 + 3 x + x^2 + 2 x^3 + 2 x^4 + 2 x^5 + x^7)/((1 + x) (1 - x + x^2) (1 + x + x^2) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 08 2014 *)

vector(60, n, n--; n+floor(n^2/2 +n^3/3)) \\ G. C. Greubel, Aug 12 2018

G.f.: x*(1+3*x+x^2+2*x^3+2*x^4+2*x^5+x^7) / ((1+x)*(1-x+x^2)*(1+x+x^2)*(1-x)^4).
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-6) -3*a(n-7) +3*a(n-8) -a(n-9).
Also, for h>=0:
a(6h)   = 6*h*( 12*h^2 + 3*h + 1 ),
a(6h+1) = 72*h^3 + 54*h^2 + 18*h + 1,
a(6h+2) = 6*( 4*h + 1 )*( 3*h^2 + 3*h + 1 ),
a(6h+3) = 2*( 36*h^3 + 63*h^2 + 39*h + 8 ),
a(6h+4) = 3*( 24*h^3 + 54*h^2 + 42*h + 11 ),
a(6h+5) = 72*h^3 + 198*h^2 + 186*h + 59.

cp's OEIS Frontend

A382488 The number of unitary 3-smooth divisors of n.

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