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

%I A064723 #58 Jul 17 2025 14:48:32
%S A064723 1,1,2,4,18,40,210,492,2786,39650,97108,1459960,9030450,22542396,
%T A064723 141358274,2249412290,36259245522,91815545800,1500020153484,
%U A064723 9702063416738,24704432285040,409634464205812,2672366681180466,44720842390302450,1927655270098608960
%N A064723 (L(p)-1)/p where L() are the Lucas numbers (A000032) and p runs through the primes.
%H A064723 Harry J. Smith, <a href="/A064723/b064723.txt">Table of n, a(n) for n=0..100</a>
%H A064723 Larry Ericksen, <a href="http://siauliaims.su.lt/index.php?option=com_content&amp;view=article&amp;id=44&amp;Itemid=9">Primality Testing and Prime Constellations</a>, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.
%H A064723 S. Litsyn and V. Shevelev, <a href="http://dx.doi.org/10.1142/S1793042105000339">Irrational Factors Satisfying the Little Fermat Theorem</a>, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
%H A064723 V. Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-March/012750.html">A property of n-bonacci constant</a>, Seqfan (Mar 23 2014)
%F A064723 a(n) = A006206(A000040(n+1)). - _Creighton Dement_, Nov 04 2005
%F A064723 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
%e A064723 a(0) = (Lucas(2) - 1)/2 = (3 - 1)/2 = 1; a(3) = (Lucas(7) - 1)/7 = (29 - 1)/7 = 4.
%p A064723 A064723 := proc(n)
%p A064723     p := ithprime(1+n) ;
%p A064723     (A000032(p)-1)/p ;
%p A064723 end proc: # _R. J. Mathar_, Jan 09 2017
%t A064723 Array[(LucasL@ Prime@ # - 1)/Prime@ # &, {23}] (* _Michael De Vlieger_, Aug 22 2015 *)
%o A064723 (PARI) lucas(n) = if(n==0,2, if(n==1,1,fibonacci(n+1)+fibonacci(n-1)))
%o A064723 forprime(n=1,100,print1((lucas(n)-1)/n, ", "))
%o A064723 (PARI) lucas(n)= { if(n==0, 2, if(n==1, 1, fibonacci(n + 1) + fibonacci(n - 1))) }
%o A064723 { n=-1; forprime (p=2, prime(101), write("b064723.txt", n++, " ", (lucas(p) - 1)/p) ) } \\ _Harry J. Smith_, Sep 23 2009
%o A064723 (Magma) [(Lucas(NthPrime(n))-1)/NthPrime(n): n in [1..40]]; // _Vincenzo Librandi_, Aug 22 2015
%Y A064723 Cf. A000032, A006206.
%K A064723 nonn
%O A064723 0,3
%A A064723 _Shane Findley_, Oct 13 2001
%E A064723 More terms from _James Sellers_ and _Klaus Brockhaus_, Oct 16 2001