A237835 -id:A237835 - cp's OEIS frontend

A237517 Pisano period of n^2 divided by Pisano period of n.

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 1, 13, 7, 15, 16, 17, 9, 19, 10, 21, 11, 23, 4, 25, 13, 27, 7, 29, 5, 31, 32, 33, 17, 35, 9, 37, 19, 39, 40, 41, 7, 43, 44, 45, 23, 47, 16, 49, 25, 17, 26, 53, 27, 55, 14, 19, 29, 59, 5, 61, 31, 63, 64, 65, 11, 67, 34, 23

Offset: 1

Charles R Greathouse IV, Feb 13 2014

nonn

For all n, a(n) | n.

Conjecture (Saha & Karthik): a(n) = 1 only for n = 1, 6, and 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Arpan Saha and C. S. Karthik, A few equivalences of Wall-Sun-Sun prime conjecture, arXiv:1102.1636 [math.NT], 2011.

Cf. A001175, A001176, A237835.

pp[1] = 1; pp[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k+1], n] == 1, Return[k]]];
a[n_] := pp[n^2]/pp[n];
Array[a, 100] (* Jean-François Alcover, Dec 04 2018 *)

fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
entry_p(p)=my(k=1,c=Mod(1,p),o); while(c,[o,c]=[c,c+o];k++); k
entry(n)=if(n==1,return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i,1]>1e14,entry_p(f[i,1]^f[i,2]), entry_p(f[i,1])*f[i,1]^(f[i,2] - 1))); if(f[1,1]==2&&f[1,2]>1, v[1]=3<

A271782 Smallest n-Wall-Sun-Sun prime.

Original entry on oeis.org

13, 241, 2, 3, 191, 5, 2, 3, 2683

Offset: 2

Felix Fröhlich, Apr 18 2016

A prime p is a k-Wall-Sun-Sun prime iff p^2 divides F_k(pi_k(p)), where F_k(n) is the k-Fibonacci numbers, i.e., a Lucas sequence of first kind with (P,Q) = (k,-1), and pi_k(p) is the Pisano period of k-Fibonacci numbers modulo p (cf. A001175, A175181-A175185).
For prime p > 2 not dividing k^2 + 4, it is a k-Wall-Sun-Sun prime iff p^2 | F_k(p-((k^2+4)/p)), where ((k^2+4)/p) is the Kronecker symbol.
a(1) would be the smallest Wall-Sun-Sun prime whose existence is an open question.
a(12)..a(16) = 2, 3, 3, 29, 2. a(18)..a(33) = 3, 11, 2, 23, 3, 3, 2, 5, 79, 3, 2, 7, 23, 3, 2, 239. a(36)..a(38) = 2, 7, 17. a(40), a(41) = 2, 3. a(43)..a(46) = 5, 2, 3, 41. - R. J. Mathar, Apr 22 2016
a(17) = 1192625911, a(35) = 153794959, a(39) = 30132289567, a(47)..a(50) = 139703, 2, 3, 3. If they exist, a(11), a(34), a(42) are greater than 10^12. - Max Alekseyev, Apr 26 2016
Column k = 1 of table T(n, k) of k-th n-Wall-Sun-Sun prime (that table is not yet in the OEIS as a sequence). - Felix Fröhlich, Apr 25 2016
From Richard N. Smith, Jul 16 2019: (Start)
a(n) = 2 if and only if n is divisible by 4.
a(n) = 3 if and only if n == 5, 9, 13, 14, 18, 22, 23, 27, 31 (mod 36). (End)

Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41:1 (2009), 497-504.
Richard N. Smith, Table of n, a(n) for n = 1..1024 (or 0 if unknown, searched up to 2^22)
Richard N. Smith, List of odd n-Wall-Sun-Sun primes <= 2^20 for 1<=n<=1024
Wikipedia, Wall-Sun-Sun prime

Cf. A001177, A039951, A113649, A113650, A113651, A214028, A237517, A237835, A241014, A244801, A253247, A268478.

A271782(k) = forprime(p=2,10^8, if( (([0,1;1,k]*Mod(1,p^2))^(p-kronecker(k^2+4,p)))[1,2]==0, return(p);); ); \\ Max Alekseyev, Apr 22 2016, corrected by Richard N. Smith, Jul 16 2019 to include p=2 and p divides k^2+4

a(4n) = 2.

Edited by Max Alekseyev, Apr 25 2016

A237836 Pisano period of n^2.

Original entry on oeis.org

1, 6, 24, 24, 100, 24, 112, 96, 216, 300, 110, 24, 364, 336, 600, 384, 612, 216, 342, 600, 336, 330, 1104, 96, 2500, 1092, 1944, 336, 406, 600, 930, 1536, 1320, 612, 2800, 216, 2812, 342, 2184, 2400, 1640, 336, 3784, 1320, 5400, 1104, 1504, 384, 5488, 7500

Offset: 1

Charles R Greathouse IV, Feb 13 2014

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001175, A001176, A237517, A237835.

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
entry_p(p)=my(k=1, c=Mod(1, p), o); while(c, [o, c]=[c, c+o]; k++); k
entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, entry_p(f[i, 1]^f[i, 2]), entry_p(f[i, 1])*f[i, 1]^(f[i, 2] - 1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<

a(n) = A001175(n^2).

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A237517 Pisano period of n^2 divided by Pisano period of n.

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A271782 Smallest n-Wall-Sun-Sun prime.

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A237836 Pisano period of n^2.

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