A064723 (L(p)-1)/p where L() are the Lucas numbers (A000032) and p runs through the primes.

1, 1, 2, 4, 18, 40, 210, 492, 2786, 39650, 97108, 1459960, 9030450, 22542396, 141358274, 2249412290, 36259245522, 91815545800, 1500020153484, 9702063416738, 24704432285040, 409634464205812, 2672366681180466, 44720842390302450, 1927655270098608960

Offset: 0

Author

Shane Findley, Oct 13 2001

Keywords

• nonn

Examples

			a(0) = (Lucas(2) - 1)/2 = (3 - 1)/2 = 1; a(3) = (Lucas(7) - 1)/7 = (29 - 1)/7 = 4.
		

Links

• Harry J. Smith, Table of n, a(n) for n=0..100
• Larry Ericksen, Primality Testing and Prime Constellations, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.
• S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
• V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)

Crossrefs

Cf. A000032, A006206.

Programs

Magma

[(Lucas(NthPrime(n))-1)/NthPrime(n): n in [1..40]]; // Vincenzo Librandi, Aug 22 2015

Maple

A064723 := proc(n)
    p := ithprime(1+n) ;
    (A000032(p)-1)/p ;
end proc: # R. J. Mathar, Jan 09 2017

Mathematica

Array[(LucasL@ Prime@ # - 1)/Prime@ # &, {23}] (* Michael De Vlieger, Aug 22 2015 *)

PARI

lucas(n) = if(n==0,2, if(n==1,1,fibonacci(n+1)+fibonacci(n-1)))
forprime(n=1,100,print1((lucas(n)-1)/n, ", "))

PARI

lucas(n)= { if(n==0, 2, if(n==1, 1, fibonacci(n + 1) + fibonacci(n - 1))) }
{ n=-1; forprime (p=2, prime(101), write("b064723.txt", n++, " ", (lucas(p) - 1)/p) ) } \\ Harry J. Smith, Sep 23 2009

Formula

a(n) = A006206(A000040(n+1)). - Creighton Dement, Nov 04 2005

a(n) = (round(phi^prime(n+1)) - 1)/prime(n+1), where phi is golden ratio (A001622). Indeed, L(p) = round(phi^p), and round(phi^p) == 1 (mod p) and, what is more, for p>=5, round(phi^p) == 1 (mod 2*p) (see Shevelev link). In particular, all terms >=2 are even. - Vladimir Shevelev, Mar 24 2014

Extensions

More terms from James Sellers and Klaus Brockhaus, Oct 16 2001