A007659 -id:A007659 - cp's OEIS frontend

A037948 Duplicate of A007659.

2, 3, 5, 7, 2411

Offset: 1

dead

A063938 Numbers k that divide tau(k), where tau(k)=A000594(k) is Ramanujan's tau function.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 88, 90, 91, 92, 96, 98, 100, 105, 108, 112, 115, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 161, 162, 168

Offset: 1

Robert G. Wilson v, Aug 31 2001

Although most small numbers are in the sequence, it becomes sparser for larger values; e.g., only 504 numbers up to 10000 and only 184 numbers from 10001 to 20000 are in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Tau Function.

For the sequence when n is prime see A007659.

Cf. A063940, A000594, A079334, A296991, A296993.

(* First do <Michael De Vlieger, Dec 23 2017 *)

for (n=1,1000,if(Mod(ramanujantau(n),n)==0,print1(n", "))) \\ Dana Jacobsen, Sep 06 2015

use ntheory ":all"; my @p = grep { !(ramanujan_tau($) % $) } 1..1000; say "@p"; # Dana Jacobsen, Sep 06 2015

from itertools import count, islice
from sympy import divisor_sigma
def A063938_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not -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, count(max(startvalue,1)))
A063938_list = list(islice(A063938_gen(),25)) # Chai Wah Wu, Nov 08 2022

More terms from Dean Hickerson, Jan 03 2003

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

A299171 Primes p such that Ramanujan number tau(p) is divisible by p+1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 97, 107, 127, 137, 139, 149, 167, 179, 191, 199, 223, 229, 239, 251, 269, 293, 349, 359, 367, 383, 419, 431, 449, 479, 499, 503, 587, 593, 599, 643, 647, 719, 809, 827, 839, 863, 881, 919

Offset: 1

Seiichi Manyama, Feb 04 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..900

Cf. A000594, A007659, A299157, A299163, A299172.

Select[Range[1000], PrimeQ[#] && Divisible[RamanujanTau[#], #+1] &] (* Amiram Eldar, Apr 14 2021 *)

isok(p) = isprime(p) && !(ramanujantau(p) % (p+1)); \\ Michel Marcus, Feb 05 2018

A299172 Primes p such that Ramanujan number tau(p) is divisible by p-1.

Original entry on oeis.org

2, 3, 59, 107

Offset: 1

Seiichi Manyama, Feb 04 2018

a(5) > 10^7.

Cf. A000594, A007659, A299171, A299204, A299205.

Select[ Prime@ Range@ 30, Mod[ RamanujanTau@#, # - 1] == 0 &] (* Robert G. Wilson v, Feb 11 2018 *)

isok(p) = isprime(p) && !(ramanujantau(p) % (p-1)); \\ Michel Marcus, Feb 05 2018

A063940 Composite numbers k such that Ramanujan's function tau(k) (A000594) is not divisible by k.

Original entry on oeis.org

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 116, 117, 118, 119, 121, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 143, 145, 146, 148, 152, 153

Offset: 1

Robert G. Wilson v, Aug 31 2001

nonn

			22 is a term because Ramanujan's tau(22) = 18643272 and 18643272 mod 22 = 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000594, A063938, A007659.

Select[ Range[ 70 ], Mod[ CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 70} ] ], 70 ], x ][ [ # ] ], # ] != 0 && ! PrimeQ[ # ] & ]
(* First do *) <Dean Hickerson, Jan 03 2003 *)

A135430 Numbers k for which Ramanujan's function tau(k)=A000594(k) is an odd prime.

Original entry on oeis.org

63001, 458329, 942841, 966289, 1510441, 2961841, 4879681, 14280841, 29019769, 46117681, 49182169, 51652969, 56957209, 75047569, 80120401, 86136961, 93644329, 97752769, 104509729, 162384049, 164378041, 177235969, 193571569

Offset: 1

Giovanni Resta, Dec 12 2007

nonn

Here, negative integers whose absolute value is prime are considered prime.
a(1) = 63001 was found by Lehmer in 1965. It is known that tau(n) is odd if and only if n is an odd square. Indeed, a(1)=251^2, a(2)=677^2, ..., a(7)=47^4. The first sixth power in the sequence is 1151^6.
From Olivier Rozier, Feb 03 2016 (Start)
a(n) = p^(q-1) for p,q odd primes, and p not included in A007659, so that a(n) is a subsequence of A036454. Consequence of the arithmetical properties: (i) tau function is multiplicative, (ii) for p prime, tau(p^(k-1)) is the k-th term of a Lucas sequence.
It is conjectured that the equation |tau(n)|=2 has no solution. (End)

			tau(63001) = -80561663527802406257321747 which is prime.

Dana Jacobsen, Table of n, a(n) for n = 1..1000
Michael Bennett, Adela Gherga, Vandita Patel, and Samir Siksek, Odd values of the Ramanujan tau function, arXiv:2101.02933 [math.NT], 2021.
D. H. Lehmer, The Primality of Ramanujan's Tau-Function, The American Mathematical Monthly, Vol. 72, No. 2, Part 2 (Feb., 1965), pp. 15-18.
N. Lygeros and O. Rozier, Odd prime values of the Ramanujan tau function, Ramanujan Journal, Vol. 32 (2013), pp. 269-280.
Eric Weisstein's World of Mathematics, Tau Function Prime

Cf. A000594 (Ramanujan's tau function tau(n)).

Cf. A265913 (tau(a(n))).

Select[Range[1, 7000, 2]^2, PrimeQ@RamanujanTau@# &]

for(x=1,1000, n=(2*x+1)^2; if(isprime(abs(ramanujantau(n))), print1(n", "))) \\ Dana Jacobsen, Sep 07 2015

use ntheory ":all"; for (0..1000) { my $n = (2*$+1)**2; say $n if is_prime(abs(ramanujan_tau($n))); } # _Dana Jacobsen, Sep 07 2015

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

A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 7, 7, 1, 5, 10, 6, 11, 12, 20, 24, 14, 12, 3, 19, 6, 37, 20, 33, 20, 27, 50, 34, 36, 29, 18, 64, 4, 2, 66, 32, 3, 64, 61, 51, 60, 84, 95, 83, 63, 97, 42, 28, 61, 67, 32, 10, 29, 73, 37, 92, 16, 120, 31, 107, 120, 141, 145, 39, 12, 74, 150

Offset: 1

Felix Fröhlich, Dec 21 2020

nonn

These are essentially the values that can be used to define "near-misses" in a search of terms for A007659, similar to how "near-Wieferich primes", "near-Wilson primes" and "near-Wall-Sun-Sun primes" are defined in searches for Wieferich primes (A001220), Wilson primes (A007540) and Wall-Sun-Sun (Fibonacci-Wieferich) primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p), Journal of Integer Sequences 13 (2010), Article 10.7.4.

Cf. A001220, A007540, A007659, A295645.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme).

a[n_] := Module[{p = Prime[n]}, Min[Abs[Mod[RamanujanTau[p], {-p, p}]]]]; Array[a, 100] (* Amiram Eldar, Jan 10 2025 *)

a(n) = my(p=prime(n)); abs(centerlift(Mod(ramanujantau(p), p)))

a(n) = 0 iff prime(n) is a term of A007659.

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

A037948 Duplicate of A007659.

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A063938 Numbers k that divide tau(k), where tau(k)=A000594(k) is Ramanujan's tau function.

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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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A299171 Primes p such that Ramanujan number tau(p) is divisible by p+1.

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A299172 Primes p such that Ramanujan number tau(p) is divisible by p-1.

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A063940 Composite numbers k such that Ramanujan's function tau(k) (A000594) is not divisible by k.

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A135430 Numbers k for which Ramanujan's function tau(k)=A000594(k) is an odd prime.

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

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