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Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.
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%I #26 Jan 09 2025 02:05:31

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

%H Amiram Eldar, <a href="/A296992/b296992.txt">Table of n, a(n) for n = 2..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TauFunction.html">Tau Function</a>.

%e tau(2) = -24 and 2^3 divides 24, so a(2) = 3.

%e tau(3) = 252 and 3^2 divides 252, so a(3) = 2.

%e tau(4) = -1472 and 4^3 divides 1472, so a(4) = 3.

%t 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 a[n_] := IntegerExponent[RamanujanTau[n], n]; Array[a, 100, 2] (* _Amiram Eldar_, Jan 09 2025 *)

%o (PARI) a(n) = valuation(ramanujantau(n), n); \\ _Amiram Eldar_, Jan 09 2025

%Y Cf. A063938 (a(n)>=1), A296991 (a(n)>=2), A296993 (a(n)>=3).

%Y Cf. A191599 (a(n)=0), A297000 (a(n)=1), A297001 (a(n)=2).

%K nonn

%O 2,1

%A _Seiichi Manyama_, Dec 22 2017