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

1, 11, 23, 691, 5807, 85583, 189751, 37264081

Offset: 1

Author

Seiichi Manyama, Nov 25 2017

Keywords

• nonn
• more

Comments

Compare with A063938.

a(9) > 8*10^7. - Seiichi Manyama, Jan 01 2018

Examples

			tau(11) = 534612 and 11 | (534612 - 1).
tau(23) = 18643272 and 23 | (18643272 - 1).
tau(691) = -2747313442193908 and 691 | (-2747313442193908 - 1).
tau(5807) = 237456233554906855056 and 5807 | (237456233554906855056 + 1).
tau(85583) = 90954516543892718450139576 and 85583 | (90954516543892718450139576 - 1).
tau(189751) = 4685230754227867924094547904 and 189751 | (4685230754227867924094547904 + 1).
tau(37264081) = 831105005803795341334403814220760726696052 and 37264081 | (831105005803795341334403814220760726696052 - 1).
		

Links

• Eric Weisstein's World of Mathematics, Tau Function.
• Wikipedia, Ramanujan tau function.

Crossrefs

Cf. A000594, A063938, A273650, A295645.

Programs

Mathematica

fQ[n_] := Block[{t = RamanujanTau@n}, Mod[t, n] == 1 || Mod[t, n] + 1 == n]; (* Robert G. Wilson v, Nov 25 2017 *)

Python

from itertools import count, islice
from sympy import divisor_sigma
def A295654_gen(startvalue=1): # generator of terms >= startvalue
    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)))
A295654_list = list(islice(A295654_gen(),4)) # Chai Wah Wu, Nov 08 2022

Formula

A273650(a(n)) is 1 or n - 1.

Extensions

a(8) from Seiichi Manyama, Jan 01 2018