A180367 a(n) = 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

Author

Jonathan Vos Post, Aug 31 2010

Keywords

• nonn

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..82 (terms 0..50 from Harvey P. Dale)

Crossrefs

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

Programs

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 *)

Formula

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

Extensions

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