A047650 -id:A047650 - cp's OEIS frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

Original entry on oeis.org

211, 281, 421, 461, 521, 691, 881, 991, 1031, 1151, 1511, 1601, 1871, 1951, 2221, 2591, 3001, 3251, 3571, 3851, 4021, 4391, 4441, 4481, 4621, 4651, 4691, 4751, 4871, 5081, 5281, 5381, 5531, 5591, 5641, 5801, 5881, 6011, 6101, 6211, 6271, 6491, 6841

Offset: 1

N. J. A. Sloane

From A.H.M. Smeets, Nov 15 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
In E. Lehmer, Artiads characterized, she counted in the table on p. 122 the primes p for which p == 1 (mod 5) instead of all primes. As a result, in the corollary on p. 121, the 20% becomes 5% (or 1/20 instead of 1/5). (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..24903 (first 1000 terms from T. D. Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).
Michael J. Mossinghoff and Christopher Pinner, Prime power order circulant determinants, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
H. W. Lloyd Tanner, On the Binomial Equation x^p-1=0: Quinquisection, Proc. London Math. Soc., 18 (1886-1887), 214-234.
H. W. Lloyd Tanner, On Complex Primes formed with the Fifth Roots of Unity, Proc. London Math. Soc., 24 (1892-1893), 223-262.

Cf. A047650, A000045, A024894, subsequence of A030430.

See also A270798 (a subsequence), A270800.

a001583 n = a001583_list !! (n-1)
a001583_list = filter
   (\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list
-- Reinhard Zumkeller, Aug 15 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* Jean-François Alcover, Jun 22 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
list(lim)=my(v=List()); forprime(p=11,lim, if(p%5==1 && fibmod(p\5,p)==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 06 2017

From A.H.M. Smeets, Nov 15 2023: (Start)
Equals {prime(m): A296240(m) == 0 (mod 5)}.
a(n) ~ prime(20*n). (End)

More terms from James Sellers, Jan 25 2000

Edited by N. J. A. Sloane, Apr 01 2016

A005968 Sum of cubes of first n Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 10, 37, 162, 674, 2871, 12132, 51436, 217811, 922780, 3908764, 16558101, 70140734, 297121734, 1258626537, 5331629710, 22585142414, 95672204155, 405273951280, 1716768021816, 7272346018247, 30806152127640, 130496954475672, 552793970116297, 2341672834801754

Offset: 0

N. J. A. Sloane

From Alexander Adamchuk, Aug 07 2006: (Start)
The only two prime terms are a(2) = 2 and a(4) = 37.
The prime p divides a(p-1) iff p is in A045468.
The prime p divides a((p-1)/2) iff p is in A047650.
3^4 divides a(p) iff p is in A003628.
3^5 divides a(p) for p = {37,53,109,181,197,269,397,431,541,...}.
3^6 divides a(p) for p = {109,541,...}.
3^7 divides a(p) for p = {557,...}. (End)

Art Benjamin, Timothy A. Carnes, and Benoit Cloitre, Recounting the Sums of Cubes of Fibonacci Numbers, Congressus Numerantium, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, (William Webb, ed.), Vol 194, pp. 45-51, 2009.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Art Benjamin, Timothy A. Carnes, and Benoit Cloitre, Counting the Sums of Cubes of Fibonacci Numbers.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
David Treeby, Hidden Formulas in Fibonacci Tilings, Fibonacci Quart. 54 (2016), no. 1, 23-30.
Index entries for linear recurrences with constant coefficients, signature (4,3,-9,2,1).

Partial sums of A056570.  Cf. A119284 (alternating sum).
Cf. A045468, A047650, A003628.
Sums of other powers: A000071, A001654, A005969, A098531, A098532, A098533, A128697.

[(1/10)*( Fibonacci(3*n+2)-(-1)^(n)*6*Fibonacci(n-1)+5 ): n in [0..30]]; // G. C. Greubel, Jan 17 2018

with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^3; printf(`%d,`,l[i]) od: # James Sellers, May 29 2000
A005968:=(-1+2*z+z**2)/(z-1)/(z**2+4*z-1)/(z**2-z-1); # conjectured by Simon Plouffe in his 1992 dissertation

f[n_]:=(Fibonacci[n]*Fibonacci[n+1]^2+(-1)^(n-1)*Fibonacci[n-1]+1)/2;Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2010 *)
Accumulate[Fibonacci[Range[0,20]]^3]
CoefficientList[Series[x*(1-2*x-x^2)/((1-x)*(1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

a(n)=(fibonacci(n)*fibonacci(n+1)^2+(-1)^(n-1)*fibonacci(n-1)+1)/2

a(n)=(fibonacci(3*n+2)-(-1)^(n)*6*fibonacci(n-1)+5)/10

PARI
```
a(n)=sum(i=1,n,fibonacci(i)^3)
```

G.f.: x*(1-2*x-x^2)/((1-x)*(1+x-x^2)*(1-4*x-x^2)). - Ralf Stephan, Apr 23 2004
a(n) = (1/2)*(F(n)*F(n+1)^2 + (-1)^(n-1)*F(n-1) + 1). - Benoit Cloitre, Aug 06 2004
a(n) = Sum_{i=1..n} A000045(i)^3.
a(n) = (1/10)*(F(3*n+2) - (-1)^(n)*6*F(n-1) + 5). - Art Benjamin and Timothy A. Carnes
a(n+5) = 4*a(n+4) + 3*a(n+3) - 9*a(n+2) + 2*a(n+1) + a(n). - Benoit Cloitre, Sep 12 2004

More terms from James Sellers, May 29 2000

A098531 Sum of fifth powers of first n Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 34, 277, 3402, 36170, 407463, 4491564, 49926988, 553211363, 6137270812, 68054635036, 754774491429, 8370420537086, 92830050637086, 1029498223070793, 11417322172518550, 126619992693837974, 1404237451180502875, 15573231068749231000

Offset: 0

Benoit Cloitre, Sep 12 2004

Prime p divides a((p-1)/2) for p = {29,89,101,181,229,...} = A047650[n]. Primes for which golden mean tau is a quadratic residue or Primes of the form x^2 + 20y^2. - Alexander Adamchuk, Aug 07 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,32,-100,20,48,-7,-1).

Cf. A000071, A001654, A005968, A005969, A047650, A056572, A098532, A098533, A119286, A128697.

[(&+[Fibonacci(k)^5:k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 17 2018

Accumulate[Fibonacci[Range[0,20]]^5]  (* Harvey P. Dale, Jan 14 2011 *)
CoefficientList[Series[x*(1-7*x-16*x^2+7*x^3+x^4)/((1-x)*(1+4*x-x^2)*(1-x-x^2)*(1-11*x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 13 2012 *)

PARI
```
a(n)=sum(i=0,n,fibonacci(i)^5)
```

a(n) = -7/22 + 2*F(n+2)/5 + (F(5*(n+1)) + F(5*n))/(5*55) - (-1)^n*(F(3*(n+1)) - F(3*n))/(2*10), where F=A000045. One may use F(5*(n+1)) + F(5*n) = F(5*n+1) + 4*F(5*n+2) (due to the Binet-de Moivre formula).

G.f.: x*(1-7*x-16*x^2+7*x^3+x^4)/((1-x)*(1+4*x-x^2)*(1-x-x^2)*(1-11*x-x^2)). - Bruno Berselli, Oct 12 2012

Formula corrected, with the author's consent, by Wolfdieter Lang, Oct 12 2012

A047652 Primes for which golden mean is a cubic residue.

Original entry on oeis.org

139, 151, 199, 331, 541, 619, 661, 709, 811, 829, 919, 1069, 1231, 1279, 1291, 1381, 1471, 1579, 1699, 1999, 2161, 2221, 2239, 2251, 2281, 2371, 2389, 2521, 2659, 2689, 2749, 3001, 3121, 3271, 3331, 3391, 3499, 3529, 3571, 3631, 3919, 4021, 4051, 4159

Offset: 1

N. J. A. Sloane

Primes of the form x^2 + xy + 34y^2, whose discriminant is -135. - T. D. Noe, May 17 2005

Primes of the form x^2 + 135*y^2. - Arkadiusz Wesolowski, May 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

Cf. A047650.

Select[Prime[Range[1000]],IntegerQ[Fibonacci[(#1-1)/3]/#1]&] (* Alexander Adamchuk, Sep 16 2006 *)

Primes p that divide Fibonacci((p-1)/3). - Alexander Adamchuk, Sep 16 2006

More terms from James Sellers, Jan 25 2000

A121569 a(n) = Fibonacci((prime(n)+3)/2) - 1.

Original entry on oeis.org

1, 2, 4, 12, 20, 54, 88, 232, 986, 1596, 6764, 17710, 28656, 75024, 317810, 1346268, 2178308, 9227464, 24157816, 39088168, 165580140, 433494436, 1836311902, 12586269024, 32951280098, 53316291172, 139583862444, 225851433716

Offset: 2

Alexander Adamchuk, Aug 08 2006

nonn

p = Prime[n] divides a(n) for p = {29,89,101,181,229,349,401,461,509,521,541,709,761,769,809,...} = A047650[n] Primes for which golden mean tau is a quadratic residue or Primes of the form x^2+20y^2.

Cf. A000045, A121567, A121568, A033205, A045468, A064739, A047650.

Table[Fibonacci[(Prime[n]+3)/2]-1,{n,2,50}]

a(n) = Fibonacci[ (Prime[n]+3)/2 ] - 1, n>1. a(n) = Sum[ Fibonacci[k], {k,1,(p-1)/2} ], p = Prime[n], n>1.

A366951 a(n) = 2(p_n - 1)/A060305(n) iff p_n == +/- 1 (mod 5), 2(p_n + 1)/A060305(n) iff p_n == +/- 2 (mod 5), 0 iff p_n = 5.

Original entry on oeis.org

2, 1, 0, 1, 2, 1, 1, 2, 1, 4, 2, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 3, 2, 3, 1, 2, 1, 6, 2, 6, 1, 1, 1, 1, 2, 4, 2, 1, 1, 18, 10, 1, 1, 4, 9, 2, 2, 2, 1, 3, 2, 2, 1, 10, 1, 1, 7, 2, 1, 1, 6, 1, 3, 4, 3, 2, 1, 1, 2, 1, 2, 1, 4, 2, 2, 10, 2, 1, 2, 1

Offset: 1

A.H.M. Smeets, Oct 29 2023

nonn

Cf. A000040, A003147, A060305, A047650, A047652, A071774, A124096, A124253, A296240, A308784 - A308794.

a(n) == 0 (mod 2) for prime(n) == +/- 1 (mod 5) and n > 2.
a(n) == 1 (mod 2) for Prime(n) == +/- 2 (mod 5) and n > 2.
a(n) = 1 iff prime(n) in A071774.
a(n) = 2 iff prime(n) in ({2} union A003147)/{5}.
a(n) = 3 iff prime(n) in A308784.
a(n) = 4 iff prime(n) in A308787.
a(n) = 6 iff prime(n) in A308788.
a(n) = 7 iff prime(n) in A308785.
a(n) = 8 iff prime(n) in A308789.
a(n) = 9 iff prime(n) in A308786.
a(n) = 10 iff prime(n) in A308790.
a(n) = 12 iff prime(n) in A308791.
a(n) = 14 iff prime(n) in A308792.
a(n) = 16 iff prime(n) in A308793.
a(n) = 18 iff prime(n) in A308794.
a(n) = A296240(n) iff prime(n) == +/- 2 (mod 5) and n > 3.
a(n) = 2*A296240(n) iff prime(n) == +/- 1 (mod 5) and n > 3.
a(n) in {2^k: k > 1} iff prime(n) in {A047650}.
a(n) == 3 (mod 6) iff prime(n) in {A124096}.
a(n) == 6 (mod 12) iff prime(n) in {A046652}.
a(n) == 0 (mod 14) iff prime(n) in {A125252}.

Showing 1-8 of 8 results.

cp's OEIS Frontend

A047651 Minimal positive value of tau modulo primes in A047650.

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A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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Mathematica

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A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

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Haskell

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A005968 Sum of cubes of first n Fibonacci numbers.

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Magma

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A098531 Sum of fifth powers of first n Fibonacci numbers.

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Magma

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A047652 Primes for which golden mean is a cubic residue.

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A121569 a(n) = Fibonacci((prime(n)+3)/2) - 1.

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A366951 a(n) = 2(p_n - 1)/A060305(n) iff p_n == +/- 1 (mod 5), 2(p_n + 1)/A060305(n) iff p_n == +/- 2 (mod 5), 0 iff p_n = 5.