A271953 a(n) is the period of A000930 modulo n.

1, 7, 8, 14, 31, 56, 57, 28, 24, 217, 60, 56, 168, 399, 248, 56, 288, 168, 381, 434, 456, 420, 528, 56, 155, 168, 72, 798, 840, 1736, 930, 112, 120, 2016, 1767, 168, 342, 2667, 168, 868, 1723, 3192, 1848, 420, 744, 3696, 46, 56, 399, 1085, 288, 168, 468, 504, 1860, 1596, 3048, 840, 3541, 1736, 1240, 6510

Offset

1,2

Links

Joerg Arndt, Table of n, a(n) for n = 1..1000

H. T. Engstrom, On sequences defined by linear recurrence relations, Trans. Am. Math. Soc. 33 (1) (1931) 210-218.

K. Kirthi, Narayana Sequences for Cryptographic Applications, arXiv preprint arXiv:1509.05745 [math.NT], 2015.

M. B. Nathanson, Linear recurrences and uniform distribution, Proc. Amer. Math. Soc. 48 (1975), 289-291.

D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

Formula

Let the prime factorization of n be p1^e1*...*pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)) [Engstrom]. - N. J. A. Sloane, Feb 18 2017

Mathematica

minlen = 100; maxlen = 2*10^4;
per[lst_] := FindTransientRepeat[lst, 2] // Last // Length;
a[n_] := Module[{p0=0, len=minlen}, While[p0 = Mod[LinearRecurrence[{1, 0, 1}, {1, 1, 1}, len], n] // per; p0<=1 && len<=maxlen, len = 2 len]; p0];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Jul 21 2018 *)

Prog

(PARI)
per(n, S, R) = {  \\ S[]: leading terms, R[]: recurrence
    if ( n==1, return( 1 ) );
    my ( r = #R );
    if ( r != #S , error("Mismatch in length of S[] and R[]") );
    S = vector(#S, j, Mod(S[j], n) );
    R = vector(#S, j, Mod(R[j], n) );
    my( T = S );
    my( j = 0 );
    until ( 0,  \\ forever
        j += 1;
        my( t = sum(i=1, r, R[i] * T[r+1-i] ) );  \\ next term
        for (k=1, r-1, T[k] = T[k+1] );
        T[r] = t;
        if ( T == S , return(j) );
    );
}
\\vector(66, n, per(n, [0, 1], [1, 1]) )  \\ A001175
\\vector(66, n, per(prime(n), [0, 1], [1, 1]) )  \\ A060305
vector(66, n, per(n, [0, 0, 1], [1, 0, 1]) )  \\ A271953
\\vector(66, n, per(prime(n), [0, 0, 1], [1, 0, 1]) )  \\ A271901
\\vector(66, n, per(n, [0, 0, 1], [0, 1, 1]) )  \\ A104217
/* Joerg Arndt, Apr 17 2016 */

Crossrefs

Cf. A000930, A271901 (periods mod primes), A001175 (periods of A000045 modulo n).

Sequence in context: A281710 A037970 A101517 * A165465 A047521 A231390

Adjacent sequences: A271950 A271951 A271952 * A271954 A271955 A271956

Keyword

nonn

Author

Joerg Arndt, Apr 17 2016

Status

approved