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

1,2

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.

Mathematica

(* First do <<NumberTheory`Ramanujan` *) test[n_] := Mod[RamanujanTau[n], n]==0; Select[Range[200], test]
(* Second program: *)
Select[Range@ 168, Divisible[RamanujanTau@ #, #] &] (* Michael De Vlieger, Dec 23 2017 *)

Prog

(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

Crossrefs

For the sequence when n is prime see A007659.

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

Sequence in context: A225737 A357315 A079333 * A002473 A174995 A161466

Adjacent sequences: A063935 A063936 A063937 * A063939 A063940 A063941

Keyword

nonn,easy

Author

Robert G. Wilson v, Aug 31 2001

Extensions

More terms from Dean Hickerson, Jan 03 2003

Status

approved