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A309583 Numbers k with 1 zero in a fundamental period of A000129 mod k.

%I A309583 #20 Jun 17 2024 10:49:57
%S A309583 1,2,4,7,8,14,16,20,23,24,28,31,32,40,41,46,47,48,49,52,56,62,64,71,
%T A309583 72,79,80,82,88,92,94,96,98,100,103,104,112,116,120,124,127,128,140,
%U A309583 142,144,148,151,152,158,160,161,164,167,168,176,184,188,191,192
%N A309583 Numbers k with 1 zero in a fundamental period of A000129 mod k.
%C A309583 Numbers k such that A214027(k) = 1.
%C A309583 The odd numbers k satisfy A175181(k) == 2 (mod 4).
%H A309583 Jianing Song, <a href="/A309583/b309583.txt">Table of n, a(n) for n = 1..5000</a>
%o A309583 (PARI) for(k=1, 200, if(A214027(k)==1, print1(k, ", ")))
%Y A309583 Cf. A175181.
%Y A309583 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 A309583                              |   m=1    |   m=2    |   m=3
%Y A309583 -----------------------------+----------+----------+---------
%Y A309583 The sequence {x(n)}          | A000045  | A000129  | A006190
%Y A309583 The sequence {w(k)}          | A001176  | A214027  | A322906
%Y A309583 Primes p such that w(p) = 1  | A112860* | A309580  | A309586
%Y A309583 Primes p such that w(p) = 2  | A053027  | A309581  | A309587
%Y A309583 Primes p such that w(p) = 4  | A053028  | A261580  | A309588
%Y A309583 Numbers k such that w(k) = 1 | A053031  | this seq | A309591
%Y A309583 Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
%Y A309583 Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
%Y A309583 * and also A053032 U {2}
%K A309583 nonn
%O A309583 1,2
%A A309583 _Jianing Song_, Aug 10 2019