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

Author

Peter Luschny, Jul 01 2020

Keywords

• nonn

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

• 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.

Crossrefs

Cf. A335947/A335948, A144845, A158302.

Programs

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)])

Formula

a(n) = min {m | m*([x^k] b(n, x)) is an 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).