A335947 -id:A335947 - cp's OEIS frontend

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

Peter Luschny, Jul 01 2020

nonn

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.

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.

Cf. A335947/A335948, A144845, A158302.

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)])

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).

A335948 T(n, k) = denominator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 240, 1, 2, 1, 1, 1, 48, 1, 6, 1, 1, 1344, 1, 16, 1, 4, 1, 1, 1, 192, 1, 48, 1, 4, 1, 1, 3840, 1, 48, 1, 24, 1, 3, 1, 1, 1, 1280, 1, 16, 1, 40, 1, 1, 1, 1, 33792, 1, 256, 1, 32, 1, 8, 1, 4, 1, 1, 1, 3072, 1, 256, 1, 32, 1, 8, 1, 12, 1, 1

Offset: 0

Peter Luschny, Jul 01 2020

See A335947 for formulas and references concerning the polynomials.

			First few polynomials are:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = -(1/12) + x^2;
b_3(x) = -(1/4)*x + x^3;
b_4(x) = (7/240) - (1/2)*x^2 + x^4;
b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;
b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;
Triangle starts:
1;
1,     1;
12,    1,    1;
1,     4,    1,   1;
240,   1,    2,   1,  1;
1,     48,   1,   6,  1,  1;
1344,  1,    16,  1,  4,  1,  1;
1,     192,  1,   48, 1,  4,  1, 1;
3840,  1,    48,  1,  24, 1,  3, 1, 1;
1,     1280, 1,   16, 1,  40, 1, 1, 1, 1;
33792, 1,    256, 1,  32, 1,  8, 1, 4, 1, 1;

Cf. A335947 (numerators), A157780 (column 0), A033469 (column 0 even indices only).

cp's OEIS Frontend

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

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