A180367 Lucas(prime(n+1)) - prime(Lucas(n)), for Lucas numbers beginning at 2.

0, 2, 6, 22, 182, 490, 3510, 9240, 63868, 1149468, 3009672, 54017304, 370246314, 969319296, 6643832358, 119218840092, 2139295466336, 5600748260454, 100501350226466, 688846502491240, 1803423556642478, 32361122671978600, 221806434537503870, 3980154972736116440

Offset

0,2

Comments

Commutator of Primes and Lucas numbers. Some subtlety in indexing -- should we start with 0th Lucas number is 2, and 0th prime is 1? As shown here, I use "first" to mean the initial value as shown in P(n) and L(n), even though their indexing differs. This is to A093062 Fibonacci(prime(i))-prime(Fibonacci(i)) as Fibonacci is to Lucas.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..50

Formula

a(n) = L(prime(n+1)) - prime(L(n)) = A000032(A000040(n+1)) - A000040(A000032(n)) = A180363(n+1) - A094894(n).

Example

a(0) = 0 because the 1st prime is 2, and the third Lucas number is A000032(2) = 3; while the 1st Lucas number is 2, and the 2nd prime is 3; with 3-3=0.
a(1) = 2 because the 2nd prime is 3, and A000032(3) = 4; while the 2nd Lucas number is 1, and the first2 prime is 2; with 4-2=2.
a(2) = 6 because the 3rd prime is 5, and the 6th Lucas number (counting "2" as first) is A000032(5) = 11; while the 3rd Lucas number is 3, and the 3rd prime is 5; with 11-5=6.
a(3) = 29 - 7 = 22. a(4) = 199 - 17 = 182.

Maple

A000032 := proc(n) option remember; if n <= 1 then op(n+1, [2, 1]) ; else procname(n-1)+procname(n-2) ; end if; end proc:
A094894 := proc(n) ithprime(A000032(n)) ; end proc:
A180363 := proc(n) A000032(ithprime(n)) ; end proc:
A180367 := proc(n) A180363(n+1)-A094894(n) ; end proc: seq(A180367(n), n=0..25) ; # R. J. Mathar, Sep 01 2010

Mathematica

Table[LucasL[Prime[n+1]]-Prime[LucasL[n]], {n, 0, 30}] (* Harvey P. Dale, Jan 01 2021 *)

Crossrefs

Cf. A000032, A000040, A094894, A180363, A093062.

Sequence in context: A171339 A032266 A095856 * A111061 A060803 A341884

Adjacent sequences: A180364 A180365 A180366 * A180368 A180369 A180370

Keyword

easy,nonn

Author

Jonathan Vos Post, Aug 31 2010

Extensions

Some indices corrected, 3 values corrected, formulas signs swapped - R. J. Mathar, Sep 01 2010

Status

approved