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A295654 Numbers k such that tau(k) +- 1 is congruent to 0 (mod k), where tau is the Ramanujan tau function (A000594).

%I A295654 #53 Feb 16 2025 08:33:52
%S A295654 1,11,23,691,5807,85583,189751,37264081
%N A295654 Numbers k such that tau(k) +- 1 is congruent to 0 (mod k), where tau is the Ramanujan tau function (A000594).
%C A295654 Compare with A063938.
%C A295654 a(9) > 8*10^7. - _Seiichi Manyama_, Jan 01 2018
%H A295654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunction.html">Tau Function</a>.
%H A295654 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan_tau_function">Ramanujan tau function</a>.
%F A295654 A273650(a(n)) is 1 or n - 1.
%e A295654 tau(11) = 534612 and 11 | (534612 - 1).
%e A295654 tau(23) = 18643272 and 23 | (18643272 - 1).
%e A295654 tau(691) = -2747313442193908 and 691 | (-2747313442193908 - 1).
%e A295654 tau(5807) = 237456233554906855056 and 5807 | (237456233554906855056 + 1).
%e A295654 tau(85583) = 90954516543892718450139576 and 85583 | (90954516543892718450139576 - 1).
%e A295654 tau(189751) = 4685230754227867924094547904 and 189751 | (4685230754227867924094547904 + 1).
%e A295654 tau(37264081) = 831105005803795341334403814220760726696052 and 37264081 | (831105005803795341334403814220760726696052 - 1).
%t A295654 fQ[n_] := Block[{t = RamanujanTau@n}, Mod[t, n] == 1 || Mod[t, n] + 1 == n]; (* _Robert G. Wilson v_, Nov 25 2017 *)
%o A295654 (Python)
%o A295654 from itertools import count, islice
%o A295654 from sympy import divisor_sigma
%o A295654 def A295654_gen(startvalue=1): # generator of terms >= startvalue
%o A295654     return filter(lambda n: n==1 or abs(-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)==1, count(max(startvalue,1)))
%o A295654 A295654_list = list(islice(A295654_gen(),4)) # _Chai Wah Wu_, Nov 08 2022
%Y A295654 Cf. A000594, A063938, A273650, A295645.
%K A295654 nonn,more
%O A295654 1,2
%A A295654 _Seiichi Manyama_, Nov 25 2017
%E A295654 a(8) from _Seiichi Manyama_, Jan 01 2018