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A056028 Numbers k such that k^18 == 1 (mod 19^2).

%I A056028 #23 Apr 08 2025 16:13:30
%S A056028 1,28,54,62,68,69,99,116,127,234,245,262,292,293,299,307,333,360,362,
%T A056028 389,415,423,429,430,460,477,488,595,606,623,653,654,660,668,694,721,
%U A056028 723,750,776,784,790,791,821,838,849,956,967,984,1014,1015,1021,1029
%N A056028 Numbers k such that k^18 == 1 (mod 19^2).
%H A056028 Matthew House, <a href="/A056028/b056028.txt">Table of n, a(n) for n = 1..10000</a>
%H A056028 <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A056028 a(n) = a(n-1) + a(n-18) - a(n-19). - _Matthew House_, Feb 12 2017
%F A056028 G.f.: x*(x^18 +27*x^17 +26*x^16 +8*x^15 +6*x^14 +x^13 +30*x^12 +17*x^11 +11*x^10 +107*x^9 +11*x^8 +17*x^7 +30*x^6 +x^5 +6*x^4 +8*x^3 +26*x^2 +27*x +1) / (x^19 -x^18 -x +1). - _Colin Barker_, Feb 12 2017
%F A056028 From _Mike Sheppard_, Feb 20 2025: (Start)
%F A056028 a(n) = a(n-18) + 19^2.
%F A056028 a(n) ~ (19^2/18)*n. (End)
%t A056028 x=19; Select[ Range[ 1250 ], PowerMod[ #, x-1, x^2 ]==1& ]
%o A056028 (PARI) isok(n) = Mod(n, 19^2)^18 == 1; \\ _Michel Marcus_, Feb 12 2017
%o A056028 (PARI) Vec(x*(x^18 +27*x^17 +26*x^16 +8*x^15 +6*x^14 +x^13 +30*x^12 +17*x^11 +11*x^10 +107*x^9 +11*x^8 +17*x^7 +30*x^6 +x^5 +6*x^4 +8*x^3 +26*x^2 +27*x +1) / (x^19 -x^18 -x +1) + O(x^100)) \\ _Colin Barker_, Feb 12 2017
%Y A056028 Cf. A056021, A056022, A056024, A056025, A056026, A056027.
%Y A056028 Cf. A381319 (general case mod n^2).
%K A056028 nonn,easy
%O A056028 1,2
%A A056028 _Robert G. Wilson v_, Jun 08 2000