A063938 Numbers k that divide tau(k), where tau(k)=A000594(k) is Ramanujan's tau function.

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

Offset: 1

Author

Robert G. Wilson v, Aug 31 2001

Keywords

• nonn
• easy

Comments

Although most small numbers are in the sequence, it becomes sparser for larger values; e.g., only 504 numbers up to 10000 and only 184 numbers from 10001 to 20000 are in the sequence.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Tau Function.

Crossrefs

For the sequence when n is prime see A007659.

Cf. A063940, A000594, A079334, A296991, A296993.

Programs

Mathematica

(* First do <Michael De Vlieger, Dec 23 2017 *)

PARI

for (n=1,1000,if(Mod(ramanujantau(n),n)==0,print1(n", "))) \\ Dana Jacobsen, Sep 06 2015

Perl

use ntheory ":all"; my @p = grep { !(ramanujan_tau($) % $) } 1..1000; say "@p"; # Dana Jacobsen, Sep 06 2015

Python

from itertools import count, islice
from sympy import divisor_sigma
def A063938_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not -840*(pow(m:=n+1>>1,2,n)*(0 if n&1 else pow(m*divisor_sigma(m),2,n))+(sum(pow(i,4,n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))<<1)) % n, count(max(startvalue,1)))
A063938_list = list(islice(A063938_gen(),25)) # Chai Wah Wu, Nov 08 2022

Extensions

More terms from Dean Hickerson, Jan 03 2003