A326577 -id:A326577 - cp's OEIS frontend

A326478 a(n) = ndenominator(nBernoulli(n-1))/denominator(Bernoulli(n-1)).

1, 1, 1, 4, 1, 6, 1, 8, 3, 10, 1, 12, 1, 14, 5, 16, 1, 18, 1, 20, 7, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 32, 11, 34, 35, 36, 1, 38, 13, 40, 1, 42, 1, 44, 3, 46, 1, 48, 7, 50, 17, 52, 1, 54, 55, 56, 19, 58, 1, 60, 1, 62, 21, 64, 13, 66, 1, 68, 23, 70, 1, 72, 1

Offset: 1

Peter Luschny, Jul 16 2019

nonn

Empirical: a(2*n) = [x^n] x*(2/(x - 1)^2 - 1) for n >= 1, implying the conjecture that a(2*n) = A103517(n+1) and/or A272651(n).

Conjectural, the odd fixed points > 1 of this sequence are A121707; in other words, for n > 1, denominator(n*Bernoulli(n-1)) = denominator(Bernoulli(n-1)) <=> n | Sum_{k=1..n-1} k^(n-1). (See the conjectures of Thomas Ordowski in A121707.)

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A326577, A326578, A103517, A272651, A027641/A027642 (Bernoulli), A121707.

A326478 := n -> n*denom(n*bernoulli(n-1))/denom(bernoulli(n-1)):
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> n/igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..73);

a[n_] := Module[{b =  BernoulliB[n - 1]}, n * Denominator[n * b] / Denominator[b]]; Array[a, 100] (* Amiram Eldar, Apr 26 2024 *)

a(n) = n*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ Michel Marcus, Jul 17 2019

a(prime(n)) = 1.

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

A326578 a(n) = n^2denominator(nBernoulli(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

Peter Luschny, Jul 16 2019

nonn

Conjecture: If n is Carmichael then a(n) = n.

Are the fixed points of this sequence the numbers satisfying Korselt's criterion?

Peter Luschny, Table of n, a(n) for n = 1..10000

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

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);

a[n_] := Module[{b =  BernoulliB[n - 1]}, n^2 * Denominator[n * b] / Denominator[b]]; Array[a, 60] (* Amiram Eldar, Apr 26 2024 *)

a(n) = n^2*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ Michel Marcus, Jul 17 2019

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.

A326579 a(n) = ndenominator(nBernoulli(n-1)) for n >= 1 and a(0) = 0.

Original entry on oeis.org

0, 1, 2, 6, 4, 30, 6, 42, 8, 90, 10, 66, 12, 2730, 14, 30, 16, 510, 18, 798, 20, 2310, 22, 138, 24, 13650, 26, 54, 28, 870, 30, 14322, 32, 5610, 34, 210, 36, 1919190, 38, 78, 40, 13530, 42, 1806, 44, 2070, 46, 282, 48, 324870, 50, 1122, 52, 1590, 54, 43890, 56

Offset: 0

Peter Luschny, Jul 17 2019

nonn

Conjecture: For n>1: denominator(Bernoulli(n-1)) = n*denominator(n*Bernoulli(n-1)) <=> n is Korselt <=> n is prime or n is Carmichael.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Math. Comp., 35 (1980), 1003-1026.
Wikipedia, Carmichael numbers, Korselt's criterion.

Cf. A326578, A326478, A326577, A027641/A027642 (Bernoulli), A002997 (Carmichael), A324050 (Korselt).

A326579 := n -> `if`(n = 0, 0, n*denom(n*bernoulli(n-1))): seq(A326579(n), n=0..56);

a[n_] := n * Denominator[n * BernoulliB[n - 1]]; a[0] = 0; Array[a, 60, 0] (* Amiram Eldar, Apr 26 2024 *)

a(n) = if (n, n*denominator(n*bernfrac(n-1)), 0); \\ Michel Marcus, Jul 19 2019

a(2*n) = 2*n.

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

Peter Luschny, Jul 19 2019

nonn

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).

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

Peter Luschny, Table of n, a(n) for n = 1..10000

Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).

Cf. A326578, A326478, A326579, A326577.

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);

a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
Array[a, 80] (* Jean-François Alcover, Jul 21 2019 *)

a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ Michel Marcus, Jul 19 2019

a(n) divides n, n/a(n) = A326478(n).

cp's OEIS Frontend

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

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

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A326579 a(n) = n*denominator(n*Bernoulli(n-1)) for n >= 1 and a(0) = 0.

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

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Maple

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Formula

A326478 a(n) = ndenominator(nBernoulli(n-1))/denominator(Bernoulli(n-1)).

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

A326579 a(n) = ndenominator(nBernoulli(n-1)) for n >= 1 and a(0) = 0.