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 1-3 of 3 results.

A326578 a(n) = n^2*denominator(n*Bernoulli(n-1))/denominator(Bernoulli(n-1)) = n*A326478(n).

Original entry on oeis.org

1, 2, 3, 16, 5, 36, 7, 64, 27, 100, 11, 144, 13, 196, 75, 256, 17, 324, 19, 400, 147, 484, 23, 576, 125, 676, 243, 784, 29, 900, 31, 1024, 363, 1156, 1225, 1296, 37, 1444, 507, 1600, 41, 1764, 43, 1936, 135, 2116, 47, 2304, 343, 2500, 867, 2704, 53, 2916, 3025
Offset: 1

Views

Author

Peter Luschny, Jul 16 2019

Keywords

Comments

Conjecture: If n is Carmichael then a(n) = n.
Are the fixed points of this sequence the numbers satisfying Korselt's criterion?

Links

Crossrefs

Cf. A326478, A326579, A326577, A027641/A027642 (Bernoulli), A002997 (Carmichael), A324050 (Korselt).

Programs

  • Maple
    A326578 := n -> n*A326478(n): seq(A326578(n), n=1..55);
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> n^2/igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..55);
  • Mathematica
    a[n_] := Module[{b =  BernoulliB[n - 1]}, n^2 * Denominator[n * b] / Denominator[b]]; Array[a, 60] (* Amiram Eldar, Apr 26 2024 *)
  • PARI
    a(n) = n^2*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ Michel Marcus, Jul 17 2019

Formula

a(prime(n)) = prime(n).
a(n) = n^2/gcd(n*N(n-1), D(n-1)), with N(k)/D(k) = B(k) the k-th Bernoulli number.

A326580 a(n) = (2*n+1)*denominator((2*n+1)*Bernoulli(2*n)).

Original entry on oeis.org

1, 6, 30, 42, 90, 66, 2730, 30, 510, 798, 2310, 138, 13650, 54, 870, 14322, 5610, 210, 1919190, 78, 13530, 1806, 2070, 282, 324870, 1122, 1590, 43890, 16530, 354, 56786730, 126, 6630, 64722, 690, 4686, 140100870, 150, 2310, 3318, 6210270, 498, 57873270, 174
Offset: 0

Views

Author

Peter Luschny, Jul 17 2019

Keywords

Links

Crossrefs

Cf. A326579, A326578, A326478, A027641/A027642 (Bernoulli).

Programs

  • Maple
    A326580 := n -> (2*n+1)*denom((2*n+1)*bernoulli(2*n)):
seq(A326580(n), n=0..43);
  • Mathematica
    Table[(2n+1)Denominator[(2n+1)BernoulliB[2n]],{n,0,50}] (* Harvey P. Dale, Jul 31 2021 *)
  • PARI
    a(n) = (2*n+1)*denominator((2*n+1)*bernfrac(2*n)); \\ Michel Marcus, Jul 19 2019

Formula

a(n) = A326579(2*n + 1).

A326584 a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 3, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 3, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 1, 3, 1, 1, 1, 37, 1, 3, 1, 41, 1, 43, 1, 15, 1, 47, 1, 7, 1, 3, 1, 53, 1, 1, 1, 3, 1, 59, 1, 61, 1, 3, 1, 5, 1, 67, 1, 3, 1, 71, 1, 73, 1, 3, 1, 1, 1, 79, 1
Offset: 1

Views

Author

Peter Luschny, Jul 19 2019

Keywords

Comments

Conjectures:
(1) If n > 1 then a(n) = n <=> n is prime or Carmichael (A002997).
(2) If n is odd then a(n) = 1 <=> n = 1 or is a term of A121707.
(3) The fixed points of n^2/a(n) are exactly the numbers satisfying Korselt's criterion (compare A326578 and A324050).

Examples

			a(559) =   1 and 559 is in A121707.
a(561) = 561 and 561 is Carmichael.
a(563) = 563 and 563 is prime.

Links

Crossrefs

Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).
Cf. A326578, A326478, A326579, A326577.

Programs

  • Maple
    db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..80);
  • Mathematica
    a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
Array[a, 80] (* Jean-François Alcover, Jul 21 2019 *)
  • PARI
    a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ Michel Marcus, Jul 19 2019

Formula

a(n) divides n, n/a(n) = A326478(n).
Showing 1-3 of 3 results.

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