Tau Function Prime

Author

Eric W. Weisstein
December 18, 2015

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/TauFunctionPrime.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TauFunctionPrime.html.

©2015 Wolfram Research, Inc. except for portions noted otherwise

Primes

Slow but check everything

taus=RamanujanTau[Range[1*^5]];//Timing

Position[taus,_?PrimeQ]//Flatten//Timing

Primes must be odd squares:

Select[Range[1,Sqrt[1*^6],2]^2,PrimeQ[RamanujanTau[#]]&]//Timing

Select[Range[1,Sqrt[1*^9],2]^2,PrimeQ[RamanujanTau[#]]&]//Timing

A135430

oeis={63001, 458329, 942841, 966289, 1510441, 2961841, 4879681, 14280841, 29019769, 46117681, 49182169, 51652969, 56957209, 75047569, 80120401, 86136961, 93644329, 97752769, 104509729, 162384049, 164378041, 177235969, 193571569};

A265913

A265913=RamanujanTau/@oeis

<<Utilities`EISFormat`

BFile["~/b265913.txt",A265913]

PrimeQ/@%

Large probable primes

http://www.primenumbers.net/prptop/searchform.php?form=Tau%28%3F%29&action=Search

largeprimes={Inactive[643^11688],Inactive[971^15738],Inactive[509^18946],Inactive[631^32370],Inactive[773^34960]};

Print[{#,IntegerLength[n=Activate[#]],PrimeQ[RamanujanTau[n]]//Timing}]&/@largeprimes;