A099856 -id:A099856 - cp's OEIS frontend

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

1, 6, 18, 54, 162, 486, 1458, 1926, 4374, 5778, 13122, 17334, 39366, 52002, 118098, 156006, 206082, 354294, 468018, 618246, 1062882, 1404054, 1854738, 2471058, 3188646, 4212162, 5564214, 7413174, 9565938, 12636486, 16692642, 22050774

Offset: 1

Robert G. Wilson v

nonn

Note that if n divides A000032(n) and p is an odd prime divisor of A000032(n), then pn divides A000032(pn) and, furthermore, p^k*n divides A000032(p^k*n) for every integer k>=0.
In particular, since 6 divides A000032(6) = 2*3^2, A016089 includes all terms of the geometric progression 2*3^k for k>0 (see A099856); since 18 divides A000032(18) = 2*3^3*107, A016089 includes all terms of the form 2*107^m*3^k for k>1 and m>=0; etc.
Terms of A016089 starting with 18 are multiples of 18. There are no other terms of the form 18p where p is prime, except for p=3 and p=107. - Alexander Adamchuk, May 11 2007

Lars Blomberg, Table of n, a(n) for n = 1..91
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000032, A000204, A025192, A008776.

Cf. A099856, A072378 = numbers n such that 12n divides Fibonacci(12n), A023172 = numbers n such that n divides Fibonacci(n).

a = 1; b = 3; Do[c = a + b; a = b; b = c; If[Mod[c, n] == 0, Print[n]], {n, 3, 2, 10^6}]

is(n)=(Mod([0,1;1,1],n)^n*[2;1])[1,1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Extended and revised by Max Alekseyev, May 13 2007, May 15 2008, May 16 2008

A099858 A Chebyshev transform of (1+3x)/(1-3x).

Original entry on oeis.org

1, 6, 17, 42, 109, 288, 755, 1974, 5167, 13530, 35423, 92736, 242785, 635622, 1664081, 4356618, 11405773, 29860704, 78176339, 204668310, 535828591, 1402817466, 3672623807, 9615053952, 25172538049, 65902560198, 172535142545

Offset: 0

Paul Barry, Oct 28 2004

The g.f. is related to the g.f. of A099856 by the Chebyshev mapping G(x)-> (1/(1+x^2))*G(x/(1+x^2)).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Cf. A099856, A099857.

LinearRecurrence[{3,-2,3,-1},{1,6,17,42},40] (* Harvey P. Dale, Apr 17 2024 *)

G.f.: (1+3*x+x^2)/((1+x^2)*(1-3*x+x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(6*3^(n-2*k-1)-0^(n-2*k)).
a(n) = Sum_{k=0..n} (0^k+6*Fibonacci(2*k))*cos(Pi*(n-k)/2).
a(n) = Sum_{k=0..n} A099857(k)*cos(Pi*(n-k)/2).
a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3)-a(n-4).
a(n) = (1/2)*(4*Fibonacci(2*n+2) - i^n - (-i)^n). - Ralf Stephan, Dec 04 2004

A164907 a(n) = (3*3^n-(-1)^n)/2.

Original entry on oeis.org

1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Klaus Brockhaus, Aug 31 2009

Interleaving of A096053 and A083884 without initial term 1.
Partial sums are (essentially) in A080926.
First differences are (essentially) in A105723.
a(n)+a(n+1) = A008776(n+1) = A099856(n+1) = A110593(n+2).
Binomial transform of A056450. Inverse binomial transform of A164908.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Equals A046717 without initial term 1 and A080925 without initial term 0. Equals A084182 / 2 from second term onward.

Cf. A096053, A083884, A080926, A105723, A008776, A099856, A110593, A056450, A164908.

Magma
```
[ (3*3^n-(-1)^n)/2: n in [0..25] ];
```

A164907:=n->(3*3^n - (-1)^n)/2; seq(A164907(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Table[(3*3^n - (-1)^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 21 2014 *)
LinearRecurrence[{2,3},{1,5},50] (* Harvey P. Dale, Oct 31 2018 *)

a(n) = 2*a(n-1)+3*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1+3*x)/((1+x)*(1-3*x)).
a(n) = 3*a(n-1)+2*(-1)^n. - Carmine Suriano, Mar 21 2014

A099857 Expansion of (1+3x+x^2)/(1-3x+x^2).

Original entry on oeis.org

1, 6, 18, 48, 126, 330, 864, 2262, 5922, 15504, 40590, 106266, 278208, 728358, 1906866, 4992240, 13069854, 34217322, 89582112, 234529014, 614004930, 1607485776, 4208452398, 11017871418, 28845161856, 75517614150, 197707680594

Offset: 0

Paul Barry, Oct 28 2004

Hankel transform is 1, -18, -36, 0, 0, 0, 0, 0, 0, 0, 0, ... - Philippe Deléham, Dec 15 2011

Michael De Vlieger, Table of n, a(n) for n = 0..2391
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (3, -1).

Cf. A000045, A099856, A099858.

CoefficientList[Series[(1+3x+x^2)/(1-3x+x^2),{x,0,30}],x] (* Harvey P. Dale, May 23 2015 *)

a(n) = 0^n + 6*Fibonacci(2*n).

A236967 Expansion of (1+3x)^2/(1-3x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 18 2023

nonn

			The prime factorization of 360 is 2*2*2*3*3*5, with factors greater than the least 3*3*5, so a(360) = 3.

Positions of 0's are A000961.
Positions of numbers > 0 are A024619.
Positions of first appearances appear to be A099856.
For "less than greatest" instead of "greater than least" we have A325226.
For multiplicities instead of parts we have A363131.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A047966 counts uniform partitions, ranks A072774.
A363128 counts partitions with more than one non-mode, complement A363129.
Cf. A001221, A001222, A052126, A053585, A061395, A064989, A071178, A105441, A307517, A325230.

Table[PrimeOmega[n]-If[n==1,0,FactorInteger[n][[1,2]]],{n,30}]

a(n) = A001222(n) - A067029(n).

a(n) = A001222(A028234(n)).

cp's OEIS Frontend

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

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A099858 A Chebyshev transform of (1+3x)/(1-3x).

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A164907 a(n) = (3*3^n-(-1)^n)/2.

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A099857 Expansion of (1+3x+x^2)/(1-3x+x^2).

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A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

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Magma

Formula

Extensions

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

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Crossrefs

Programs

Mathematica

Formula

A236967 Expansion of (1+3x)^2/(1-3x)^2.