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A052462 a(n) is the minimal positive integral solution k to 24*k == 1 (mod 5^n).

%I A052462 #47 Aug 11 2025 07:04:13
%S A052462 4,24,99,599,2474,14974,61849,374349,1546224,9358724,38655599,
%T A052462 233968099,966389974,5849202474,24159749349,146230061849,603993733724,
%U A052462 3655751546224,15099843343099,91393788655599,377496083577474
%N A052462 a(n) is the minimal positive integral solution k to 24*k == 1 (mod 5^n).
%C A052462 Related to a Ramanujan congruence for the partition function P = A000041.
%C A052462 Extending work of Ramanujan, Watson (1938) proved that P(m) == 0 (mod 5^n) if 24*m == 1 (mod 5^n). In particular, P(a(n)) == 0 (mod 5^n). - _Petros Hadjicostas_, Jul 29 2020
%H A052462 Vincenzo Librandi, <a href="/A052462/b052462.txt">Table of n, a(n) for n = 1..1000</a>
%H A052462 G. N. Watson, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0179">Ramanujans Vermutung über Zerfällungsanzahlen</a>, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%H A052462 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionPCongruences.html">Partition Function P Congruences</a>.
%H A052462 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,25,-25).
%F A052462 G.f.: x*(-25*x^2 + 20*x + 4)/((1 - x)*(1 - 5*x)*(1 + 5*x)).
%F A052462 a(n) = (1 + (21 + 2*(-1)^n)*5^n)/24.  - _Bruno Berselli_, Apr 04 2011
%F A052462 a(n) = a(n-1) + 25*a(n-2) - 25*a(n-3). - _Vincenzo Librandi_, Jul 01 2012
%F A052462 A000041(a(n)) == 0 (mod 5^n). - _Petros Hadjicostas_, Jul 29 2020
%e A052462 From _Petros Hadjicostas_, Jul 29 2020: (Start)
%e A052462 A000041(a(1)) = A000041(4) = 5 == 0 (mod 5).
%e A052462 A000041(a(2)) = A000041(24) = 1575 == 0 (mod 5^2).
%e A052462 A000041(a(3)) = A000041(99) = 169229875 == 0 (mod 5^3).
%e A052462 A000041(a(4)) = A000041(599) = 435350207840317348270000 == 0 (mod 5^4). (End)
%t A052462 Table[PowerMod[24, -1, 5^a], {a, 21}]
%t A052462 CoefficientList[Series[(-25x^2+20x+4)/((1-x)(1-5x)(1+5x)),{x,0,30}],x] (* _Vincenzo Librandi_, Jul 01 2012 *)
%o A052462 (Magma) I:=[4, 24, 99]; [n le 3 select I[n] else Self(n-1)+25*Self(n-2)-25*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jul 01 2012
%o A052462 (PARI) a(n) = lift(Mod(24, 5^n)^-1) \\ _David A. Corneth_ and _Petros Hadjicostas_, Jul 29 2020
%Y A052462 Cf. A000041, A052463, A052465, A052466.
%K A052462 nonn,easy
%O A052462 1,1
%A A052462 _Eric W. Weisstein_
%E A052462 Name edited by _Petros Hadjicostas_, Jul 29 2020