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A309591 Numbers k with 1 zero in a fundamental period of A006190 mod k.

%I A309591 #20 Jun 17 2024 10:50:18
%S A309591 1,2,3,4,6,9,12,18,23,27,36,43,46,53,54,61,69,79,81,86,92,101,103,106,
%T A309591 107,108,122,127,129,131,138,139,158,159,162,172,173,179,183,191,199,
%U A309591 202,206,207,211,212,214,237,243,244,251,254,258,262,263,276
%N A309591 Numbers k with 1 zero in a fundamental period of A006190 mod k.
%C A309591 Numbers k such that A322906(k) = 1.
%C A309591 The odd numbers k satisfy A175182(k) == 2 (mod 4).
%H A309591 Jianing Song, <a href="/A309591/b309591.txt">Table of n, a(n) for n = 1..3000</a>
%o A309591 (PARI) for(k=1, 300, if(A322906(k)==1, print1(k, ", ")))
%Y A309591 Cf. A175182.
%Y A309591 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 A309591                              |   m=1    |   m=2   |   m=3
%Y A309591 -----------------------------+----------+---------+----------
%Y A309591 The sequence {x(n)}          | A000045  | A000129 | A006190
%Y A309591 The sequence {w(k)}          | A001176  | A214027 | A322906
%Y A309591 Primes p such that w(p) = 1  | A112860* | A309580 | A309586
%Y A309591 Primes p such that w(p) = 2  | A053027  | A309581 | A309587
%Y A309591 Primes p such that w(p) = 4  | A053028  | A261580 | A309588
%Y A309591 Numbers k such that w(k) = 1 | A053031  | A309583 | this seq
%Y A309591 Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
%Y A309591 Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
%Y A309591 * and also A053032 U {2}
%K A309591 nonn
%O A309591 1,2
%A A309591 _Jianing Song_, Aug 10 2019