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A052466 a(n) is the smallest positive solution k to 24*k == 1 (mod 13^n).

%I A052466 #50 Feb 16 2025 08:32:42
%S A052466 6,162,1007,27371,170176,4625692,28759737,781741941,4860395546,
%T A052466 132114388022,821406847267,22327331575711,138817757188116,
%U A052466 3773319036295152,23460200964791597,637690917133880681,3964773963049779886,107769764995625835082,670046799755412800727
%N A052466 a(n) is the smallest positive solution k to 24*k == 1 (mod 13^n).
%C A052466 Related to a generalization of a Ramanujan congruence for the partition function P = A000041.
%C A052466 Atkin and O'Brien (1967) proved that for all integral n >= 1, there is an integral constant K(n) not divisible by 13 s.t. P(169*m - 7) == K(n)*P(m) (mod 13^n) for all integral m >= 1 that satisfy 24*m == 1 (mod 13^n). In particular, P(169*a(n) - 7) == K(n)*P(a(n)) (mod 13^n) for all n >= 1. Unfortunately, the calculation of the integral constants K(n) depends on several recursions found in the paper. (For each n, there are infinitely many such K(n)'s, but one may choose the smallest one that satisfies the above property.) See Theorem 2, p. 444, in their paper, even though their P is different that the P = A000041 here. - _Petros Hadjicostas_, Jul 29 2020
%C A052466 From _Petros Hadjicostas_, Aug 02 2020: (Start)
%C A052466 Assume n = 2*m, where m >= 1, and 24*k == 1 (mod 13^(2*m)), where k >= 1. Then there is an integer x = x(k) s.t. 24*k - 1 = 169^m*x. Then 1 = 24*k - 169^m*x == 0 - 1^m*x == -x (mod 24). With x = x(k) = 23, we find a(2*m), the smallest value of k >= 1 that satisfies 24*k == 1 (mod 13^(2*m)). Thus, a(2*m) = (1 + 23*13^(2*m))/24.
%C A052466 Assume now n = 2*m + 1, where m >= 0, and 24*k == 1 (mod 13^(2*m+1)), where k >= 1. Then there is an integer x = x(k) s.t. 24*k - 1 = 13*169^m*x. Then 1 = 24*k - 13*169^m*x == 0 - 13*1^m*x == -13*x (mod 24). With x = x(k) = 11, we find a(2*m+1), the smallest value of k >= 1 that satisfies 24*k == 1 (mod 13^(2*m)). Thus, a(2*m+1) = (1 + 11*13^(2*m+1))/24. (End)
%H A052466 Vincenzo Librandi, <a href="/A052466/b052466.txt">Table of n, a(n) for n = 1..900</a>
%H A052466 A. O. L. Atkin and J. N. O'Brien, <a href="https://doi.org/10.1090/S0002-9947-1967-0214540-7">Some Properties of p(n) and c(n) Modulo Powers of 13</a>, Trans. Amer. Math. Soc. 126 (1967), 442-459.
%H A052466 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionPCongruences.html">Partition Function P Congruences</a>.
%H A052466 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,169,-169).
%F A052466 G.f.: x*(-169*x^2 + 156*x + 6)/((1 - x)*(1 - 13*x)*(1 + 13*x)). - _Vincenzo Librandi_, Jul 01 2012
%F A052466 a(n) = a(n-1) + 169*a(n-2) - 169*a(n-3). - _Vincenzo Librandi_, Jul 01 2012
%F A052466 From _Petros Hadjicostas_, Aug 02 2020: (Start)
%F A052466 a(n) = (1 + 11*13^n)/24, if n is odd, and a(n) = (1 + 23*13^n)/24, if n is even.
%F A052466 a(n) - a(n-1) = 12*13^(n-1) for n even >= 2, and 5*13^(n-1) for n odd >= 3. (End)
%e A052466 From _Petros Hadjicostas_, Jul 29 2020: (Start)
%e A052466 The only value of the constant K(n) that appears explicitly in Atkin and O'Brien (1967) is K(2) = 45 (see p. 453). We then have
%e A052466 P(169*a(2) - 7) - K(2)*P(a(2)) = P(169*162 - 7) - 45*P(162) = A000041(27371) - 45*A000041(162) = A000041(27371) - 5846125708665 == 0 (mod 13^2).
%e A052466 Thus, we must have A000041(27371) == 99 (mod 169). (End)
%t A052466 Table[PowerMod[24, -1, 13^d], {d, 20}]
%t A052466 CoefficientList[Series[(-169x^2+156x+6)/((1-x)(1-13x)(1+13x)),{x,0,40}],x] (* _Vincenzo Librandi_, Jul 01 2012 *)
%t A052466 LinearRecurrence[{1,169,-169},{6,162,1007},30] (* _Harvey P. Dale_, Mar 15 2015 *)
%o A052466 (Magma) I:=[6, 162, 1007]; [n le 3 select I[n] else Self(n-1)+169*Self(n-2)-169*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 01 2012
%o A052466 (PARI) a(n) = lift(Mod(24, 13^n)^-1) \\ _Petros Hadjicostas_, Jul 29 2020
%o A052466 (SageMath)
%o A052466 def a(n): return 24.inverse_mod(13^n)
%o A052466 print([a(n) for n in range(1, 20)]) # _Peter Luschny_, Jul 30 2020
%Y A052466 Cf. A000041, A052462, A052463, A052465.
%K A052466 nonn,easy
%O A052466 1,1
%A A052466 _Eric W. Weisstein_
%E A052466 Name edited by _Petros Hadjicostas_, Jul 29 2020