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A296992 Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.

%I A296992 #27 Feb 16 2025 08:33:52
%S A296992 3,2,3,1,3,1,3,2,1,0,2,0,1,1,3,0,2,0,1,2,0,0,3,1,0,2,1,0,1,0,3,0,0,1,
%T A296992 2,0,0,0,2,0,2,0,0,1,0,0,2,1,1,0,0,0,2,0,1,0,0,0,1,0,0,1,3,0,0,0,0,0,
%U A296992 1,0,2,0,0,1,0,0,0,0,1,2,0,0,2,0,0,0,1,0,1,1,1,0,0
%N A296992 Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.
%H A296992 Amiram Eldar, <a href="/A296992/b296992.txt">Table of n, a(n) for n = 2..10000</a>
%H A296992 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunction.html">Tau Function</a>.
%e A296992 tau(2) =   -24 and 2^3 divides   24, so a(2) = 3.
%e A296992 tau(3) =   252 and 3^2 divides  252, so a(3) = 2.
%e A296992 tau(4) = -1472 and 4^3 divides 1472, so a(4) = 3.
%t A296992 f[n_] := Block[{m = 0}, While[Mod[RamanujanTau@n, n^m] == 0, m++]; m - 1]; Array[f, 93, 2] (* _Robert G. Wilson v_, Dec 23 2017 *)
%t A296992 a[n_] := IntegerExponent[RamanujanTau[n], n]; Array[a, 100, 2] (* _Amiram Eldar_, Jan 09 2025 *)
%o A296992 (PARI) a(n) = valuation(ramanujantau(n), n); \\ _Amiram Eldar_, Jan 09 2025
%Y A296992 Cf. A063938 (a(n)>=1), A296991 (a(n)>=2), A296993 (a(n)>=3).
%Y A296992 Cf. A191599 (a(n)=0), A297000 (a(n)=1), A297001 (a(n)=2).
%K A296992 nonn
%O A296992 2,1
%A A296992 _Seiichi Manyama_, Dec 22 2017