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A296991 Numbers k such that k^2 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

%I A296991 #28 Feb 16 2025 08:33:52
%S A296991 1,2,3,4,6,8,9,12,16,18,21,24,27,32,36,40,42,48,54,64,72,81,84,96,108,
%T A296991 120,128,135,144,162,168,189,192,216,243,256,270,280,288,324,336,360,
%U A296991 378,384,432,448,486,512,540,576,640,648,672,729,756,768,828,840,864
%N A296991 Numbers k such that k^2 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.
%C A296991 2^k is a term for k >= 0.
%H A296991 Seiichi Manyama, <a href="/A296991/b296991.txt">Table of n, a(n) for n = 1..500</a>
%H A296991 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunction.html">Tau Function</a>
%t A296991 fQ[n_] := Mod[RamanujanTau@n, n^2] == 0; Select[Range@875, fQ] (* _Robert G. Wilson v_, Dec 23 2017 *)
%o A296991 (PARI) is(n) = Mod(ramanujantau(n), n^2)==0 \\ _Felix Fröhlich_, Dec 24 2017
%o A296991 (Python)
%o A296991 from itertools import count, islice
%o A296991 from sympy import divisor_sigma
%o A296991 def A296991_gen(startvalue=1): # generator of terms >= startvalue
%o A296991     return filter(lambda n: not -24*((m:=n+1>>1)**2*(0 if n&1 else m*(35*m - 52*n)*divisor_sigma(m)**2)+sum(i**3*(70*i - 140*n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) % n**2, count(max(startvalue,1)))
%o A296991 A296991_list = list(islice(A296991_gen(),20)) # _Chai Wah Wu_, Nov 08 2022
%Y A296991 Cf. A000594, A063938, A296992, A296993.
%K A296991 nonn
%O A296991 1,2
%A A296991 _Seiichi Manyama_, Dec 22 2017