A295654 -id:A295654 - cp's OEIS frontend

A273650 a(n) = A000594(n) mod n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 10, 0, 7, 0, 0, 20, 1, 0, 0, 16, 0, 0, 24, 0, 21, 0, 21, 32, 0, 0, 31, 22, 27, 0, 30, 0, 31, 24, 0, 22, 27, 0, 0, 0, 21, 28, 29, 0, 45, 0, 54, 4, 14, 0, 49, 54, 0, 0, 30, 24, 64, 36, 45, 0, 19, 0, 67, 70, 0, 32, 42, 54, 37, 0, 0, 18

Offset: 1

Seiichi Manyama, May 27 2016

nonn

			tau(10) mod 10 = (-115920) mod 10 = 0,
tau(11) mod 11 = 534612 mod 11 = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000594, A063938, A295654.

a[n_] := Mod[RamanujanTau[n], n]; Array[a, 100] (* Amiram Eldar, Jan 08 2025 *)

a(n)=ramanujantau(n)%n \\ assumes the GRH; Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A273650(n): return -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 # Chai Wah Wu, Nov 08 2022

a(n) = A000594(n) mod n.
From Amiram Eldar, Jan 08 2025: (Start)
a(A063938(n)) = 0.
abs(a(A295654(n))) = 1. (End)

A295645 Primes p such that tau(p) +- 1 is congruent to 0 (mod p), where tau is the Ramanujan tau function (A000594).

Original entry on oeis.org

11, 23, 691, 5807

Offset: 1

Seiichi Manyama, Nov 25 2017

Nik Lygeros and Olivier Rozier found a new solution to the equation tau(p) + 1 == 0 (mod p) for prime p = 692881373, on September 6 2009. - Seiichi Manyama, Dec 30 2017
a(5) > 8*10^7. - Seiichi Manyama, Jan 01 2018
A superset of A193855. - Jud McCranie, Nov 06 2020

			tau(11) = 534612 and 11 | (534612 - 1), so a(1) = 11.
tau(23) = 18643272 and 23 | (18643272 - 1), so a(2) = 23.
tau(691) = -2747313442193908 and 691 | (-2747313442193908 - 1), so a(3) = 691.
tau(5807) = 237456233554906855056 and 5807 | (237456233554906855056 + 1), so a(4) = 5807.

N. Lygeros and O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.
Wikipedia, Ramanujan tau function.

Cf. A000594, A076847 (tau(p)), A007659, A193855, A273650, A295654.

Select[Prime@ Range[10^3], Function[p, AnyTrue[RamanujanTau[p] + {-1, 1}, Divisible[#, p] &]]] (* Michael De Vlieger, Dec 30 2017 *)

isok(p) = my(rp=ramanujantau(p)); isprime(p) && !((rp-1) % p) || !((rp+1) % p); \\ Michel Marcus, Nov 07 2020

cp's OEIS Frontend

A273650 a(n) = A000594(n) mod n.

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