A064723 -id:A064723 - cp's OEIS frontend

A180363 a(n) = Lucas(prime(n)).

3, 4, 11, 29, 199, 521, 3571, 9349, 64079, 1149851, 3010349, 54018521, 370248451, 969323029, 6643838879, 119218851371, 2139295485799, 5600748293801, 100501350283429, 688846502588399, 1803423556807921, 32361122672259149, 221806434537978679

Offset: 1

Jonathan Vos Post, Aug 31 2010

This is to A030426, Fibonacci(prime(n)), as A000032 (Lucas numbers beginning at 2) is to A000045.

			a(1) = 3 because the 1st prime is 2, and the 2nd Lucas number is A000032(2) = 3.
a(2) = 4 because the 2nd prime is 3, and the 3rd Lucas number is A000032(3) = 4.
a(3) = 11 because the 3rd prime is 5, and the 5th Lucas number is A000032(5) = 11.

Table of n, a(n) for n = 1..650
A. Aksenov, The Newman phenomenon and Lucas sequence, arXiv:1108.5352 [math.NT], 2011. [Gives factorizations of first 88 terms]
Paula Burkhardt et al., Visual properties of generalized Kloosterman sums, arXiv:1505.00018 [math.NT], 2015.

Cf. A000032, A000040, A000045, A030426.

[Lucas(NthPrime(n)): n in [1..30]]; // Vincenzo Librandi, Dec 01 2015

A180363 := proc(n) A000032(ithprime(n)) ; end proc: seq(A180363(n),n=1..30) ; # R. J. Mathar, Sep 01 2010
# second Maple program:
a:= n-> (<<1|1>, <1|0>>^ithprime(n). <<2, -1>>)[1, 1]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 03 2022

LucasL[Prime[Range[30]]] (* Vincenzo Librandi, Dec 01 2015 *)

from sympy import lucas, prime
def a(n): return lucas(prime(n))
print([a(n) for n in range(1, 24)]) # Michael S. Branicky, Dec 30 2021

a(n) = A000032(A000040(n)) = Lucas(prime(n)).

a(n) = A032170(A000040(n)) / A064723(n-1) - 1 for n>1. - Flávio V. Fernandes, Dec 30 2021

Entries checked by R. J. Mathar, Sep 01 2010

Edited by N. J. A. Sloane, Nov 28 2011

A014981 a(n) = c(prime(n))/prime(n), where c = Perrin sequence A001608 (starting 0,2,3,...) and prime(n) is the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 11, 28, 120, 197, 892, 2479, 4148, 11687, 56010, 271913, 461529, 2270882, 6599404, 11263855, 56250108, 164879269, 830987861, 7231032935, 21386730355, 36802336319, 109099442316, 187943217386

Offset: 1

Clark Kimberling, Robert G. Wilson v

Seiichi Manyama, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)
Ondrej Such, An Insufficient Condition for Primality, Problem 10268, Amer. Math. Monthly, Vol. 103, No. 10 (1996), p. 911.

See A001608, the main entry for the Perrin sequence.

Cf. A000040, A052338, A064723.

c(n) = polsym(x^3-x-1,n)[n+1]; \\ A001608
a(n) = my(p=prime(n)); c(p)/p; \\ Michel Marcus, Mar 03 2022

A242926 a(n) = denominator of B(0,n), where B(n,n) = 0, B(n-1,n) = 1/n and otherwise B(m,n) = B(m-1,n+1) - B(m-1,n).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 4, 3, 5, 1, 4, 1, 7, 15, 8, 1, 18, 1, 10, 21, 11, 1, 24, 5, 13, 9, 14, 1, 30, 1, 16, 11, 17, 35, 12, 1, 19, 39, 20, 1, 42, 1, 22, 9, 23, 1, 48, 7, 25, 17, 26, 1, 54, 55, 28, 19, 29, 1, 20, 1

Offset: 0

Paul Curtz, May 26 2014

nonn

The numerators are A189731(n).
B(0,n) = 0, 1, 1, 3/2, 2, 17/6, 4, 23/4, 25/3, 61/5, 18, 107/4, 40, 421/7, ...
is a super autosequence as defined in A242563.
The positive integers in B(0,n) give A064723(n). Corresponding rank: A006093(n+1). B(0,n) is linked to the primes A000040.
Divisor of B(0,n), n > 0: 1, 1, 1, 2, 2, 4, 5, ... = A172128(n+1).
Common (LCM) denominators for the antidiagonals: 1, 1, 1, 2, 2, 6, 6, 12, 12, ... = A139550(n+1)?.
1 = 1
1/2 +  3/2 = 2
1/3 +  5/6 + 17/6 = 4
1/4 + 7/12 +  7/4 + 23/4 = 25/3
etc.
The positive terms of the first bisection are the sum of the corresponding antidiagonal terms upon the 0's.
0 followed by A001610(n), i.e., 0, 0, 2, 3, 6, 10, 17, ... is an autosequence of the second kind.

Cf. A000204, A175386, A001610, A189731, A139550, A242563.

Table[Denominator[(LucasL[n+1]-1)/(n+1)], {n, 0, 100}] (* Artur Jasinski, Nov 06 2022 *)

a(2n+1) = A175386(n).
a(n) = denominator(A001610(n)/(n+1)). [edited by Michel Marcus, Nov 14 2022]
a(n) = denominator((A000204(n+1) - 1)/(n+1)). - Artur Jasinski, Nov 06 2022

a(24)-a(60) from Jean-François Alcover, May 26 2014

A213327 Analog of Fermat quotients: a(n) = (round((phi_2)^p)-2)/p, where phi_2 is silver ratio 1+sqrt(2) and p = prime(n).

Original entry on oeis.org

2, 4, 16, 68, 1476, 7280, 189120, 986244, 27676612, 4346071600, 23696518916, 3930960079760, 120508933265760, 669708812842692, 20814182249890948, 3654563002853231440, 650000099752136709444, 3664265812073801505200, 660535426260570501876228

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2013

nonn

For similar sequence for base 2, see A007663, for a similar sequence for golden ratio, see A064723.

Cf. A007663, A096060, A146211, A180511, A164740, A222207, A064723, A213328.

A213328 Analog of Fermat quotients: a(n) = (round((phi_3)^p)-3)/p, where phi_3 = (3+sqrt(13))/2 and p = prime(n).

Original entry on oeis.org

4, 11, 78, 612, 46374, 428040, 38948910, 380144556, 37367223558, 38467601033550, 392545092308724, 426897839167539480, 45841425452161683630, 476794964068892779068, 51906117696097060014342, 59746844088106673671809870, 69664778857791165966384195366

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2013

nonn

For similar sequence for base 2, see A007663. For similar sequences for golden ratio and silver ratio, see A064723 and A213327. Note that phi_3 is called "bronze ratio".

Cf. A007663, A096060, A146211, A180511, A164740, A222207, A064723, A213327.

A096540 Quotients T(p,k)/p, where T(p,k) is the sub-triangle defined in A096539 of the triangle of coefficients of Lucas polynomials (cf. A034807).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 7, 5, 1, 1, 5, 12, 14, 7, 1, 1, 7, 26, 55, 66, 42, 12, 1, 1, 8, 35, 91, 143, 132, 66, 15, 1, 1, 10, 57, 204, 476, 728, 715, 429, 143, 22, 1, 1, 13, 100, 506, 1771, 4389, 7752, 9690, 8398, 4862, 1768, 364, 35, 1, 1, 14, 117, 650, 2530, 7084, 14421

Offset: 1

Lekraj Beedassy, Jun 24 2004

Cf. A034807, A096539.

Row sums are in A064723. - Klaus Brockhaus, May 29 2009

T(n,k)=if(k<0 || 2*k>n,0,binomial(n-k,k)+binomial(n-k-1,k-1)+(n==0 && k==0)) \\from A034807
forprime(p=2,31, for(k=1,p\2,print1(T(p,k)/p,",")))

Edited and extended by Klaus Brockhaus, Jun 27 2004

cp's OEIS Frontend

A180363 a(n) = Lucas(prime(n)).

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A014981 a(n) = c(prime(n))/prime(n), where c = Perrin sequence A001608 (starting 0,2,3,...) and prime(n) is the n-th prime.

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A242926 a(n) = denominator of B(0,n), where B(n,n) = 0, B(n-1,n) = 1/n and otherwise B(m,n) = B(m-1,n+1) - B(m-1,n).

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A213327 Analog of Fermat quotients: a(n) = (round((phi_2)^p)-2)/p, where phi_2 is silver ratio 1+sqrt(2) and p = prime(n).

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A213328 Analog of Fermat quotients: a(n) = (round((phi_3)^p)-3)/p, where phi_3 = (3+sqrt(13))/2 and p = prime(n).

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A096540 Quotients T(p,k)/p, where T(p,k) is the sub-triangle defined in A096539 of the triangle of coefficients of Lucas polynomials (cf. A034807).

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