Ramanujan's Tau function

[Eric W. Weisstein]

On Mon, 30 Dec 2002, Jon Perry wrote:

> Sequence : A000594
> 
> Most values of T(n) exhibit the property n divides |T(n)|, e.g. 3 divides
> T(3)=252
> 
> I checked with Pari, which crashes at n=17, and found n=11 (534612) and n=13
> (577738)
> 
> which break this rule.
> 
> This would make an interesting sequence - can anyone supply more terms?

Are you sure about "most"?  This seems to be the exception rather than the 
rule.

For congruence identities, see
http://mathworld.wolfram.com/TauFunction.html.

This is relatively straightforward to calculate in Mathematica.  These are 
the values of n for which the congruence Mod[tau[n],n]==0:

In[1]:=
<<NumberTheory`Ramanujan`

In[2]:=
l=Table[RamanujanTau[n],{n,3000}];//Timing

Out[2]=
{13.57 Second,Null}

In[3]:=
Position[MapIndexed[Mod[#1,#2[[1]]]&,l],0]//Flatten

Out[3]=
{1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,28,30,32,35,36,40,42,45,\
48,49,50,54,56,60,63,64,70,72,75,80,81,84,88,90,91,92,96,98,100,105,108,112,\
115,120,125,126,128,135,140,144,147,150,160,161,162,168,175,180,182,189,192,\
196,200,207,210,216,224,225,230,240,243,245,250,252,256,264,270,273,276,280,\
288,294,300,315,320,322,324,336,343,345,350,360,364,375,378,384,392,400,405,\
414,420,432,440,441,448,450,455,460,480,483,486,490,500,504,512,525,540,546,\
549,560,567,576,588,600,616,625,630,640,644,648,672,675,686,690,700,720,728,\
729,735,736,750,756,768,784,792,800,805,810,819,828,840,864,875,882,896,900,\
910,920,928,945,960,966,972,980,1000,1008,1024,1029,1035,1050,1080,1092,1098,\
1120,1125,1134,1144,1152,1176,1200,1215,1225,1250,1260,1280,1288,1296,1320,\
1323,1344,1350,1365,1372,1380,1400,1408,1440,1449,1456,1458,1470,1500,1512,\
1536,1568,1575,1600,1610,1620,1638,1656,1680,1701,1715,1728,1750,1764,1792,\
1800,1820,1840,1848,1875,1890,1920,1932,1944,1960,1975,2000,2016,2024,2025,\
2048,2058,2070,2093,2100,2160,2184,2187,2196,2200,2205,2208,2240,2250,2268,\
2275,2300,2304,2352,2376,2400,2401,2411,2415,2430,2450,2457,2484,2500,2520,\
2560,2576,2592,2625,2639,2646,2688,2700,2730,2744,2745,2760,2784,2800,2835,\
2875,2880,2898,2912,2916,2940,3000}

In[4]:=
Length[%]

Out[4]=
295


Cheers,
-E