cp's OEIS Frontend

A056021 Numbers k such that k^4 == 1 (mod 5^2).

%I A056021 #70 Mar 25 2025 16:53:39
%S A056021 1,7,18,24,26,32,43,49,51,57,68,74,76,82,93,99,101,107,118,124,126,
%T A056021 132,143,149,151,157,168,174,176,182,193,199,201,207,218,224,226,232,
%U A056021 243,249,251,257,268,274,276,282,293,299,301,307,318,324,326,332,343,349
%N A056021 Numbers k such that k^4 == 1 (mod 5^2).
%C A056021 Numbers congruent to {1, 7, 18, 24} mod 25.
%C A056021 These terms (apart from 1) are tetration bases characterized by a constant convergence speed strictly greater than 1 (see A317905). - _Marco Ripà_, Jan 25 2024
%H A056021 Amiram Eldar, <a href="/A056021/b056021.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Colin Barker)
%H A056021 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A056021 G.f.: x*(x^2+3*x+1)^2 / ((1+x)*(x^2+1)*(x-1)^2). - _R. J. Mathar_, Oct 25 2011
%F A056021 a(n) = (-25 - (-1)^n + (9-9*i)*(-i)^n + (9+9*i)*i^n + 50*n) / 8, where i = sqrt(-1). - _Colin Barker_, Oct 16 2015
%F A056021 From _Mike Sheppard_, Feb 18 2025: (Start)
%F A056021 a(n) = a(n-1) + a(n-4) - a(n-5).
%F A056021 a(n) ~ (5^2/4)*n. (End)
%t A056021 Select[ Range[ 400 ], PowerMod[ #, 4, 25 ]==1& ]
%t A056021 (* or *)
%t A056021 LinearRecurrence[{1, 0, 0, 1, -1}, {1, 7, 18, 24, 26}, 100] (* _Mike Sheppard_, Feb 18 2025 *)
%o A056021 (PARI) a(n) = (-25 - (-1)^n + (9-9*I)*(-I)^n + (9+9*I)*I^n + 50*n) / 8 \\ _Colin Barker_, Oct 16 2015
%o A056021 (PARI) Vec(x*(x^2+3*x+1)^2/((1+x)*(x^2+1)*(x-1)^2) + O(x^100)) \\ _Colin Barker_, Oct 16 2015
%o A056021 (PARI) for(n=0, 1e3, if(n^4 % 5^2 == 1, print1(n", "))) \\ _Altug Alkan_, Oct 16 2015
%o A056021 (PARI) isok(k) = Mod(k, 25)^4 == 1; \\ _Michel Marcus_, Jun 30 2021
%Y A056021 Cf. A056022, A056024, A056025, A056026, A056027, A056028, A056031, A056034, A056035, A317905.
%Y A056021 Cf. A381319 (general case mod n^2)
%K A056021 nonn,easy
%O A056021 1,2
%A A056021 _Robert G. Wilson v_, Jun 08 2000