A309592 - OEIS

%I #22 Jun 17 2024 10:50:10

%S 7,8,11,14,15,16,17,19,20,21,22,24,28,30,31,32,33,34,35,38,39,40,42,

%T 44,45,47,48,49,51,52,55,56,57,59,60,62,63,64,66,67,68,70,71,72,75,76,

%U 77,78,80,83,84,85,87,88,90,91,93,94,95,96,98,99,100,102,104

%N Numbers k with 2 zeros in a fundamental period of A006190 mod k.

%C Numbers k such that A322906(k) = 2.

%C This sequence contains all numbers k such that 4 divides A322907(k). As a consequence, this sequence contains all numbers congruent to 7, 11, 15, 19, 31, 47 modulo 52.

%C This sequence contains all odd numbers k such that 8 divides A175182(k).

%H Jianing Song, <a href="/A309592/b309592.txt">Table of n, a(n) for n = 1..10000</a>

%o (PARI) for(k=1, 100, if(A322906(k)==2, print1(k, ", ")))

%Y Cf. A175182, A322907.

%Y Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.

%Y                              |   m=1    |   m=2   |   m=3

%Y -----------------------------+----------+---------+----------

%Y The sequence {x(n)}          | A000045  | A000129 | A006190

%Y The sequence {w(k)}          | A001176  | A214027 | A322906

%Y Primes p such that w(p) = 1  | A112860* | A309580 | A309586

%Y Primes p such that w(p) = 2  | A053027  | A309581 | A309587

%Y Primes p such that w(p) = 4  | A053028  | A261580 | A309588

%Y Numbers k such that w(k) = 1 | A053031  | A309583 | A309591

%Y Numbers k such that w(k) = 2 | A053030  | A309584 | this seq

%Y Numbers k such that w(k) = 4 | A053029  | A309585 | A309593

%Y * and also A053032 U {2}

%K nonn

%O 1,1

%A _Jianing Song_, Aug 10 2019