cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-39 of 39 results.

A217036 Term preceding the first zero in the Fibonacci numbers modulo n.

Original entry on oeis.org

1, 2, 1, 3, 5, 6, 5, 8, 7, 1, 5, 8, 13, 11, 9, 4, 17, 1, 9, 13, 1, 22, 17, 18, 5, 26, 13, 1, 11, 1, 17, 23, 21, 6, 17, 31, 1, 14, 29, 40, 13, 42, 1, 26, 45, 46, 41, 48, 7, 35, 25, 23, 53, 34, 41, 20, 1, 1, 41, 11, 1, 55, 33, 47, 23, 66, 33, 22, 41, 1, 17, 27
Offset: 2

Views

Author

David Spies, Sep 24 2012

Keywords

Comments

The multiplicative order of term n modulo n is given by sequence A001176.
Let M = [{1, 1}, {1, 0}], I = [{1, 0}, {0, 1}] is the 2 X 2 identity matrix, then A001177(n) is the smallest k > 0 such that M^k == r*I (mod n) for some r such that 0 <= r < n, and a(n) gives the value r. - Jianing Song, Jul 04 2019

Links

Programs

  • Maple
    a:= proc(n) local f, g; f, g:= 1, 0;
      while f<>0 do f, g:= irem(f+g, n), f od; g
    end:
seq(a(n), n=2..100);  # Alois P. Heinz, Sep 24 2012
  • Mathematica
    Table[k = 1; While[Mod[Fibonacci[k], n] > 0, k++]; Mod[Fibonacci[k - 1], n], {n, 2, 100}] (* T. D. Noe, Sep 24 2012 *)
  • PARI
    a(n)=my(a=0,b=1);for(k=1,n^2,[a,b]=[b,(a+b)%n];if(!b,return(a))) \\ Charles R Greathouse IV, Sep 24 2012

Formula

a(n) = F(G(n)-1) mod n where G(n) is sequence A001177 and F(m) is the m-th Fibonacci number. In particular, if n is a Fibonacci number, the n-th term is the previous Fibonacci number.
From Jianing Song, Jul 04 2019: (Start)
Also a(n) = F(G(n)+1) mod n.
a(2^e) = 1 if e = 1, 2, 2^(e-1) + 1 if e >= 3; a(p^e) = a(p)^(p^(e-1)) mod p^e for odd primes p.
For odd primes p, a(p^e) = 1 if and only if A001177(p) == 2 (mod 4); a(p^e) = p^e - 1 if and only if 4 divides A001177(p). (End)

Extensions

a(14)-a(70) from Charles R Greathouse IV, Sep 24 2012

A222413 All primes p > 5 such that A001175(p) is smaller than the maximal value permitted by Wall's Theorems 6 and 7.

Original entry on oeis.org

29, 47, 89, 101, 107, 113, 139, 151, 181, 199, 211, 229, 233, 263, 281, 307, 331, 347, 349, 353, 401, 421, 461, 509, 521, 541, 557, 563, 619, 661, 677, 691, 709, 743, 761, 769, 797, 809, 811, 829, 859, 881, 911, 919, 941, 953, 967, 977, 991, 1009, 1021, 1031, 1049, 1061, 1069, 1087, 1097, 1103, 1109, 1151, 1217, 1223, 1229, 1231, 1249, 1277
Offset: 1

Views

Author

N. J. A. Sloane, Feb 28 2013

Keywords

Comments

Included because A001175 is still a mystery (as are many sequences of the same type).
A222414 gives the corresponding values of A001175(a(n)).
The maximal value for a prime p > 5 is p-1 if p == 1 or 9 (mod 10) and 2*(p+1) if p == 3 or 7 (mod 10). See Wall's Theorems 6 and 7. These values are given in A253806. - Wolfdieter Lang, Jan 16 2015
Prime(n) is a member if and only if A296240(n) > 1. - Jonathan Sondow, Dec 10 2017

Examples

			From _Wolfdieter Lang_, Jan 16 2015: (Start)
a(1) = 29 because A001175(29) = 14 but the maximal value is 29 - 1 = 28.
a(2) = 47 because A001175(47) = 32 but the maximal value is 2*(47 + 1) = 96.
All other primes p > 5 have A001175(p) = maximal value for p.
E.g., p = 11 has  A001175(11) = 11-1 = 10 and  p = 7 has A001175(7) = 2*(7 + 1) = 16. (End)

Links

Crossrefs

Cf. A001175, A060305, A222414, A296240.
Cf. A001176, A001177. - Wolfdieter Lang, Jan 16 2015

Extensions

Name corrected by Wolfdieter Lang, Jan 16 2015

A233282 Positions of records in partial LCM-products (A233287) of Fibonacci entry points (A001177).

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 17, 25, 37, 43, 47, 53, 59, 67, 73, 79, 97, 125, 127, 137, 157, 163, 179, 193, 243, 257, 269, 277, 283, 311, 313, 317, 337, 353, 359, 397, 439, 457, 479, 487, 499, 509, 523, 547, 563, 577, 613, 617, 625, 653, 673, 677, 719, 729, 733, 757, 787
Offset: 1

Views

Author

Antti Karttunen, Dec 13 2013

Keywords

Comments

Seems also to be positions of records for lcm_{i=1..n} A001175(i).
After 1, only primes and prime powers seem to occur.
Positions where a(n) is not a prime are 1, 8, 18, 25, 49, 54, 65, 75, 102, 111, 136, 164, 237, ...

Links

Crossrefs

A233283 gives the corresponding record values.
Cf. A001175, A001176, A001177, A233281, A233287.

A237835 a(n) = n*(Pisano period of n) divided by (Pisano period of n^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 4, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 4, 3, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 1
Offset: 1

Views

Author

Charles R Greathouse IV, Feb 13 2014

Keywords

Links

Crossrefs

Cf. A237517, A001175, A001176.

Programs

  • Mathematica
    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_] := n pp[n]/pp[n^2];
Array[a, 100] (* Jean-François Alcover, Dec 06 2018 *)
  • PARI
    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<

Formula

a(n) = n/A237517(n).

A235715 a(n) = number of times (n-1) occurs in the fundamental period of Fibonacci numbers modulo n.

Original entry on oeis.org

1, 2, 3, 1, 4, 6, 4, 1, 5, 8, 1, 1, 4, 8, 3, 1, 4, 4, 1, 2, 1, 2, 4, 1, 4, 8, 8, 2, 1, 6, 1, 1, 1, 4, 4, 1, 4, 1, 3, 1, 4, 2, 4, 1, 5, 4, 4, 1, 4, 8, 3, 2, 4, 8, 1, 2, 1, 2, 1, 1, 4, 1, 3, 1, 4, 2, 4, 1, 1, 8, 1, 1, 4, 8, 3, 1, 2, 6, 1, 1, 8, 8, 4, 1, 4, 8
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 17 2014

Keywords

Comments

a(n) = A128924(n,n).

Links

Crossrefs

Cf. A001176, A001175.

Programs

  • Haskell
    a235715 1 = 1
a235715 n = f 1 ps 0 where
   f 0 (1 : xs) z = z
   f _ (x : xs) z = f x xs (z + 0 ^ (n - 1 - x))
   ps = 1 : 1 : zipWith (\u v -> (u + v) `mod` n) (tail ps) ps

A386236 Ratio of the period and the reduced period of the Fibonacci 3-step sequence A000073 mod n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 3, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 3
Offset: 1

Views

Author

Peter Munn, Jul 16 2025

Keywords

Comments

The period is A046738(n) and the reduced period is A046737(n).
See also the information in A154754 and A046737.

Links

Crossrefs

Cf. A000073, A046737, A046738.
The equivalent sequence for Fibonacci numbers is A001176.
Cf. A060839 (differs first at n=31), A154754 (restriction to prime indices).

Formula

a(n) = A046738(n)/A046737(n).

A116515 a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

Original entry on oeis.org

1, 2, 4, 2, 1, 4, 4, 1, 2, 1, 1, 4, 2, 2, 2, 4, 1, 4, 2, 1, 4, 1, 2, 4, 4, 1, 2, 2, 4, 4, 2, 1, 4, 1, 4, 1, 4, 2, 2, 4, 1, 1, 1, 4, 4, 1, 1, 2, 2, 1, 4, 1, 2, 1, 4, 2, 4, 1, 4, 2, 2, 4, 2, 1, 4, 4, 1, 4, 2, 1, 4, 1, 2, 4, 1, 2, 4, 4, 2, 2, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 2, 2, 1
Offset: 1

Views

Author

Nick Krempel, Mar 24 2006

Keywords

Comments

Conditions on p_n mod 4 and mod 5 restrict possible values of a(n). The unknown (?) case is p = 1 mod 4 and (5|p) = 1, equivalently, p = 1 or 9 mod 20, where {1, 2, 4} all occur.
Number of zeros in fundamental period of Fibonacci numbers mod prime(n). [From T. D. Noe, Jan 14 2009]

Examples

			a(4) = 2, as 7 is the 4th prime, the Fibonacci numbers mod 7 have period 16, the first Fibonacci number divisible by 7 is F(8) = 21 = 3*7 and 16 / 8 = 2.
One period of the Fibonacci numbers mod 7 is 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, which has two zeros. Hence a(4)=2. [From _T. D. Noe_, Jan 14 2009]

Crossrefs

Cf. A060305, A001602.
Cf. A112860, A053027, A053028 (primes producing 1, 2 and 4 zeros) [From T. D. Noe, Jan 14 2009]

Formula

a(n) = A060305(n) / A001602(n). a(n) is always one of {1, 2, 4}.
a(n) = A001176(prime(n)) [From T. D. Noe, Jan 14 2009]

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

Views

Author

Charles R Greathouse IV, Feb 13 2014

Keywords

Links

Crossrefs

Cf. A001175, A001176, A237517, A237835.

Programs

  • PARI
    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<

Formula

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

A253808 a(n) = A106291(n)/(modified A223486(n)), n >= 1.

Original entry on oeis.org

1, 1, 4, 2, -4, 4, 4, -12, 4, -12, 2, -24, -28, 4, -8, -24, -36, 4, 2, -12, -16, 2, 4, -24, -20, -84, 4, -48, 2, -24, 2, -48, -40, -36, -16, -24, -76, 2, -56, -12, 4, -48, 4, 2, -24, 4, 4, -24, 4, -60, -72, -84, -108, 4, -20, -48, -72, 2, 2, -24, -60, 2, -48, -96, -28, -120, 4, -36
Offset: 1

Views

Author

Wolfdieter Lang, Jan 20 2015

Keywords

Comments

This sequence uses A223486 with the first two entries 1, 3, as obtained when one considers the Lucas numbers A000204 for n >= 1 (not A000032 for n >= 0).

Crossrefs

Cf. A106291, A223486, A001176 (Fibonacci case).

Formula

a(n) = A106291(n)/(modified A223486(n)), n >= 1, where one uses A223486(1) = 1 and A223486(2) = 3.
Previous Showing 31-39 of 39 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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