A335949 a(n) = denominator(b_n(x)), where b_n(x) are the polynomials defined in A335947.

1, 1, 12, 4, 240, 48, 1344, 192, 3840, 1280, 33792, 3072, 5591040, 430080, 245760, 49152, 16711680, 983040, 522977280, 27525120, 1211105280, 173015040, 1447034880, 62914560, 22900899840, 4580179968, 1409286144, 469762048, 116769423360, 4026531840, 7689065201664

Offset

0,3

Comments

The sequence can also be computed without reference to the Bernoulli polynomials (ultimately thanks to the von Staudt-Clausen theorem) by the method of Kellner and Sondow (2019). Compare the SageMath program.

Links

Table of n, a(n) for n=0..30.

Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, arXiv:1902.10672 [math.NT], 2019.

Formula

a(n) = min {m | m*([x^k] b(n, x)) is integer for all k = 0..n}.

The odd part of a(n) is squarefree (A000265).

a(n) and A144845(n) have the same odd prime factors.

a(n)/A144845(n) = 4^floor(n/2)/2 for n >= 1.

a(n)/rad(a(n)) = A158302(n+1), (rad=A007947).

Prog

(SageMath)
def A335949(n):
    a = set(prime_divisors(n + 1)) - set([2])
    b = (
        p
        for p in prime_range(3, (n + 2) // (2 + n % 2))
        if not p.divides(n + 1) and sum((n + 1).digits(base=p)) >= p
    )
    p = list(a.union(set(b)))
    return 4 ^ (n // 2) * mul(p)
print([A335949(n) for n in range(31)])

Crossrefs

Cf. A335947/A335948, A144845, A158302.

Sequence in context: A144630 A107670 A157782 * A002679 A282578 A205141

Adjacent sequences: A335946 A335947 A335948 * A335950 A335951 A335952

Keyword

nonn

Author

Peter Luschny, Jul 01 2020

Status

approved