A052462 - OEIS

%I #45 Feb 16 2025 08:32:42

%S 4,24,99,599,2474,14974,61849,374349,1546224,9358724,38655599,

%T 233968099,966389974,5849202474,24159749349,146230061849,603993733724,

%U 3655751546224,15099843343099,91393788655599,377496083577474

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

%C Related to a Ramanujan congruence for the partition function P = A000041.

%C 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 Vincenzo Librandi, <a href="/A052462/b052462.txt">Table of n, a(n) for n = 1..1000</a>

%H G. N. Watson, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002174499">Ramanujans Vermutung über Zerfällungsanzahlen</a>, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionPCongruences.html">Partition Function P Congruences</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,25,-25).

%F G.f.: x*(-25*x^2 + 20*x + 4)/((1 - x)*(1 - 5*x)*(1 + 5*x)).

%F a(n) = (1 + (21 + 2*(-1)^n)*5^n)/24.  - _Bruno Berselli_, Apr 04 2011

%F a(n) = a(n-1) + 25*a(n-2) - 25*a(n-3). - _Vincenzo Librandi_, Jul 01 2012

%F A000041(a(n)) == 0 (mod 5^n). - _Petros Hadjicostas_, Jul 29 2020

%e From _Petros Hadjicostas_, Jul 29 2020: (Start)

%e A000041(a(1)) = A000041(4) = 5 == 0 (mod 5).

%e A000041(a(2)) = A000041(24) = 1575 == 0 (mod 5^2).

%e A000041(a(3)) = A000041(99) = 169229875 == 0 (mod 5^3).

%e A000041(a(4)) = A000041(599) = 435350207840317348270000 == 0 (mod 5^4). (End)

%t Table[PowerMod[24, -1, 5^a], {a, 21}]

%t CoefficientList[Series[(-25x^2+20x+4)/((1-x)(1-5x)(1+5x)),{x,0,30}],x] (* _Vincenzo Librandi_, Jul 01 2012 *)

%o (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 (PARI) a(n) = lift(Mod(24, 5^n)^-1) \\ _David A. Corneth_ and _Petros Hadjicostas_, Jul 29 2020

%Y Cf. A000041, A052463, A052465, A052466.

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_

%E Name edited by _Petros Hadjicostas_, Jul 29 2020