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.

Showing 1-3 of 3 results.

A071774 Related to Pisano periods: integers k such that the period of Fibonacci numbers mod k equals 2*(k+1).

Original entry on oeis.org

3, 7, 13, 17, 23, 37, 43, 53, 67, 73, 83, 97, 103, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 257, 277, 283, 293, 313, 317, 337, 367, 373, 383, 397, 433, 443, 457, 463, 467, 487, 503, 523, 547, 577, 587, 593, 607, 613, 617, 643, 647, 653, 673, 683, 727
Offset: 1

Views

Author

Benoit Cloitre, Jun 04 2002

Keywords

Comments

Terms are primes with final digit 3 or 7.
Apparently these are the primes given in A003631 without 2 and A216067. - Klaus Purath, Dec 11 2020
If k is a term, then for m=5*k the period of Fibonacci numbers mod m equals 2*(m+5). - Matthew Goers, Jan 13 2021

Links

Crossrefs

Cf. A001175, A060305, A071776.
Cf. A003631, A216067.

Programs

  • Mathematica
    Select[Prime@ Range[129], Function[n, Mod[Last@ NestWhile[{Mod[#2, n], Mod[#1 + #2, n], #3 + 1} & @@ # &, {1, 1, 1}, #[[1 ;; 2]] != {0, 1} &], n] == Mod[2 (n + 1), n] ]] (* Michael De Vlieger, Mar 31 2021, after Leo C. Stein at A001175 *)
  • PARI
    for(n=2,5000,t=2*(n+1);good=1;if(fibonacci(t)%n==0, for(s=0,t,if(fibonacci(t+s)%n!=fibonacci(s)%n,good=0;break); if(s>1&&s
  • PARI
    forprime(p=3,3000,if(p%5==2||p%5==3,a=1;b=0;c=1;while(a!=0||b!=1,c++;d=a;a=b;a=(a+d)%p;b=d%p);if(c==(2*(p+1)),print1(p",")))) /* V. Raman, Nov 22 2012 */

Extensions

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 21 2004

A216067 Prime numbers p such that p is odd and is congruent to 2 (mod 5) or 3 (mod 5), but the period of the irreducible polynomial x^2-x-1 in GF(p^2) is not 2*(p+1).

Original entry on oeis.org

47, 107, 113, 233, 263, 307, 347, 353, 557, 563, 677, 743, 797, 953, 967, 977, 1087, 1097, 1103, 1217, 1223, 1277, 1307, 1427, 1483, 1523, 1553, 1597, 1733, 1823, 1877, 1913, 1973, 2027, 2207, 2237, 2243, 2267, 2333, 2417, 2447, 2663, 2687, 2753, 2777
Offset: 1

Views

Author

V. Raman, Sep 01 2012

Keywords

Examples

			47 is in the sequence because the period of the Fibonacci / Lucas numbers (mod 47) = 32, is not 2*(47+1) = 96.

Links

Crossrefs

Cf. A001175, A060305, A071776.
Cf. A071774.

Programs

  • PARI
    forprime(p=3,3000,if(p%5==2||p%5==3,a=1;b=0;c=1;while(a!=0||b!=1,c++;d=a;a=b;a=(a+d)%p;b=d%p);if(c!=(2*(p+1)),print1(p",")))) \\ V. Raman, Nov 22 2012

Extensions

Definition corrected by V. Raman, Nov 22 2012

A227397 Related to Pisano periods: Numbers k such that the period of Fibonacci numbers mod k equals k+2.

Original entry on oeis.org

4, 34, 46, 94, 106, 166, 226, 274, 334, 346, 394, 454, 514, 526, 586, 634, 694, 706, 766, 886, 934, 1006, 1126, 1174, 1186, 1234, 1294, 1306, 1354, 1366, 1486, 1546, 1654, 1714, 1726, 1774, 1894, 1954, 1966, 2026, 2326, 2374, 2386, 2434, 2566, 2614, 2734, 2746
Offset: 1

Views

Author

Matthew Goers, Sep 20 2013

Keywords

Comments

This sequence is a subsequence of A220168, where k divides the Fibonacci number F(k+2). There is no discernible pattern among the terms of A220168 terms that are not in this sequence.
All terms are 2 less than a multiple of 6, and all except the first term (4) are 2 less than a multiple of 12.

Examples

			The Pisano period (A001175) for dividing the Fibonacci numbers (A000045) by 4 is 6; 6 = 4 + 2, so 4 is a term.
The Pisano period for the Fibonacci numbers mod 34 is 36; 36 = 34 + 2, so 34 is a term.

Links

Crossrefs

Cf. A000045, A001175, A220168, A071776.
Showing 1-3 of 3 results.

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