cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A297494 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^10*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

513, 20708, 584874, 4714408, 72449100, 200562418, 1012788198, 1953009460, 6172747128, 24788658690, 37242612640, 107770200778, 198936710910, 265200653548, 449592659568, 931777815258, 1775665528380, 2155635964450, 3812897562148, 5368106367720, 6351988507678
Offset: 1

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Author

Seiichi Manyama, Dec 31 2017

Keywords

Crossrefs

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), A297493 (m=8), this sequence (m=10).

Formula

Let b(n) = 42*n^6 - 90*n^4 - 75*n^3 - 35*n^2 - 9*n - 1.
a(n) = b(prime(n)) - tau(prime(n)) where tau(n)=A000594(n) is Ramanujan's tau function.
So tau(prime(n)) + 1 == -a(n) (mod prime(n)).

A193855 Primes p such that tau(p) is congruent to 1 (mod p), where tau is the Ramanujan tau function.

Original entry on oeis.org

11, 23, 691
Offset: 1

Views

Author

Omar E. Pol, Aug 14 2011

Keywords

Comments

M. J. Hopkins wrote "It is not known whether or not tau(p) == 1 mod p holds for infinitely many primes". For more information about this open problem see the Sloane comment in A000594.
a(4) > 500000. - Dana Jacobsen, Sep 06 2015
a(4) > 10^7. - Seiichi Manyama, Nov 25 2017
Terms 23 and 691 are exceptional primes for Ramanujan's tau function, see A262339. - Jud McCranie, Nov 05 2020
A subset of A295645. - Jud McCranie, Nov 06 2020

References

  • M. J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317.
  • M. J. Hopkins, Algebraic topology and modular forms, ICM 2002, Vol. I, pp. 283-309.

Crossrefs

Programs

  • Mathematica
    Select[Prime[Range[1, 1000]], 1 == Mod[RamanujanTau[#], #] &] (* Robert Price, May 20 2015 *)
  • PARI
    forprime(n=1,1000,if(Mod(ramanujantau(n),n)==1,print1(n,", "))) \\ Dana Jacobsen, Sep 06 2015
  • Perl
    use ntheory ":all"; forprimes { say if (ramanujan_tau($) % $) == 1; } 1000; # Dana Jacobsen, Sep 06 2015
    

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

Views

Author

Seiichi Manyama, Nov 25 2017

Keywords

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).
		

Crossrefs

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

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

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Author

Felix Fröhlich, Dec 21 2020

Keywords

Comments

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.

Crossrefs

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

Programs

  • Mathematica
    a[n_] := Module[{p = Prime[n]}, Min[Abs[Mod[RamanujanTau[p], {-p, p}]]]]; Array[a, 100] (* Amiram Eldar, Jan 10 2025 *)
  • PARI
    a(n) = my(p=prime(n)); abs(centerlift(Mod(ramanujantau(p), p)))

Formula

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

Views

Author

Seiichi Manyama, Dec 16 2017

Keywords

Comments

a(4) > 10^7.
There is no odd prime p (< 10^7) such that tau(p) is congruent to (p+1)/2 (mod p).

Examples

			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.
		

Crossrefs

Showing 1-5 of 5 results.