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

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, 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, 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

Offset: 2

Author

Seiichi Manyama, Dec 22 2017

Keywords

• nonn

Examples

			tau(2) =   -24 and 2^3 divides   24, so a(2) = 3.
tau(3) =   252 and 3^2 divides  252, so a(3) = 2.
tau(4) = -1472 and 4^3 divides 1472, so a(4) = 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10000
• Eric Weisstein's World of Mathematics, Tau Function.

Crossrefs

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

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

Programs

Mathematica

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 *)
a[n_] := IntegerExponent[RamanujanTau[n], n]; Array[a, 100, 2] (* Amiram Eldar, Jan 09 2025 *)

PARI

a(n) = valuation(ramanujantau(n), n); \\ Amiram Eldar, Jan 09 2025