A273650 -id:A273650 - cp's OEIS frontend

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

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

A299163 a(n) = A000594(n) mod (n+1).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 0, 6, 7, 9, 0, 11, 0, 6, 8, 14, 0, 1, 0, 0, 2, 0, 0, 10, 15, 24, 0, 10, 0, 18, 0, 30, 12, 21, 12, 20, 30, 6, 24, 4, 0, 3, 16, 21, 0, 15, 0, 21, 43, 15, 20, 21, 0, 45, 0, 27, 42, 34, 0, 28, 46, 42, 56, 38, 48, 60, 16, 0, 14, 63, 0, 50, 60, 36, 12

Offset: 1

Seiichi Manyama, Feb 04 2018

nonn

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

Cf. A000594, A273650, A299157.

a[n_] := Mod[RamanujanTau[n], n+1]; Array[a, 100] (* Amiram Eldar, Jan 09 2025 *)

PARI
```
{a(n) = ramanujantau(n)%(n+1)}
```

A299204 a(n) = A000594(n) mod (n-1).

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 4, 5, 0, 2, 9, 2, 0, 0, 1, 2, 3, 2, 2, 12, 18, 10, 22, 7, 12, 22, 2, 2, 5, 2, 11, 16, 15, 2, 31, 2, 12, 32, 3, 2, 8, 2, 27, 42, 27, 22, 9, 9, 16, 32, 32, 10, 33, 18, 0, 0, 30, 0, 29, 2, 38, 50, 28, 20, 39, 26, 48, 48, 0, 2, 4, 2, 5, 26, 35, 12

Offset: 2

Seiichi Manyama, Feb 05 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..10000

Cf. A000594, A273650, A299163, A299172, A299205.

f[n_] := Mod[RamanujanTau@n, n - 1]; Array[f, 76, 2] (* Robert G. Wilson v, Feb 07 2018 *)

PARI
```
{a(n) = ramanujantau(n)%(n-1)}
```

A273651 a(n) = A000594(p) mod p, where p = prime(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 10, 7, 1, 24, 21, 31, 30, 31, 27, 29, 14, 49, 64, 19, 67, 37, 20, 56, 20, 74, 50, 34, 73, 29, 109, 64, 4, 137, 66, 32, 154, 64, 106, 51, 119, 97, 95, 110, 63, 102, 169, 28, 166

Offset: 1

Seiichi Manyama, May 27 2016

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Seiichi Manyama)

Cf. A000594, A007659, A273650.

Mod[RamanujanTau@ #, #] & /@ Prime@ Range@ 80 (* Michael De Vlieger, May 27 2016 *)

a(n,p=prime(n))=(65*sigma(p, 11)+691*sigma(p, 5)-691*252*sum(k=1, p-1, sigma(k, 5)*sigma(p-k, 5)))/756%p \\ Charles R Greathouse IV, Jun 07 2016

from sympy import prime, divisor_sigma
def A273651(n):
    p = prime(n)
    return -1680*sum(pow(i,4,p)*divisor_sigma(i)*divisor_sigma(p-i) for i in range(1,p+1>>1)) % p # Chai Wah Wu, Nov 08 2022

require 'prime'
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  s10 = [s1 - 1, m].min
  (0..s10).each{|i|
    s20 = [s2 - 1, m - i].min
    (0..s20).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary
end
def power(ary, n, m)
  return [1] if n == 0
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A000594(n)
  ary = Array.new(n + 1, 0)
  i = 0
  j, k = 2 * i + 1, i * (i + 1) / 2
  while k <= n
    i & 1 == 1? ary[k] = -j : ary[k] = j
    i += 1
    j, k = 2 * i + 1, i * (i + 1) / 2
  end
  power(ary, 8, n).unshift(0)[1..n]
end
def A273651(n)
  p_ary = Prime.each.take(n)
  t_ary = A000594(p_ary[-1])
  p_ary.inject([]){|s, i| s << t_ary[i - 1] % i}
end
p A273651(n)

for n > 1, a(n) = -1680*Sum_{i=1..(p-1)/2} i**4*sigma(i)*sigma(p-i) mod p where p = prime(n). - Chai Wah Wu, Nov 08 2022

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

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Nov 25 2017

Compare with A063938.

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

			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).

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

Cf. A000594, A063938, A273650, A295645.

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

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

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

a(8) from Seiichi Manyama, Jan 01 2018

A296580 Odd primes p such that tau(p) is congruent to (p-1)/2 (mod p), where tau is the Ramanujan tau function (A000594).

Original entry on oeis.org

191, 5399, 1259393

Offset: 1

Seiichi Manyama, Dec 16 2017

a(4) > 10^7.

There is no odd prime p (< 10^7) such that tau(p) is congruent to (p+1)/2 (mod p).

			tau(191) = 2762403350592 and 2762403350592 == 95 mod 191, so a(1) = 191.
tau(5399) = -616400667743946780600 and -616400667743946780600 == 2699 mod 5399, so a(2) = 5399.
tau(1259393) = -600367974333827988240021654527358 and -600367974333827988240021654527358 == 629696 mod 1259393, so a(3) = 1259393.

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

Cf. A000594, A007659, A076847 (tau(p)), A193855, A273650, A273651, A295645.

cp's OEIS Frontend

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

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A299163 a(n) = A000594(n) mod (n+1).

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A299204 a(n) = A000594(n) mod (n-1).

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A273651 a(n) = A000594(p) mod p, where p = prime(n).

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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).

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A296580 Odd primes p such that tau(p) is congruent to (p-1)/2 (mod p), where tau is the Ramanujan tau function (A000594).

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Crossrefs