Original entry on oeis.org
1, 1, 2, 4, 8, 8, 96, 192, 1152, 384, 1536, 1536, 18432, 18432, 73728, 147456, 884736, 884736, 10616832, 10616832, 212336640, 212336640, 2548039680, 849346560, 152882380800, 30576476160, 366917713920, 40768634880, 163074539520, 163074539520, 1956894474240
Offset: 0
-
def A341107(n):
def L(n, p, r):
s, q = 0, p - r
while q <= n:
s += n // q
q *= p
return s
if n < 2: return 1
p = prod(p^(L(n, p, 1) - L(n+1, p, 0)) for p in primes(n+1))
q = prod(p for p in prime_divisors(n + 1))
r = prod(p for p in (2..(n + 2)//(2 + n % 2))
if is_prime(p) and sum((n+1).digits(base = p)) >= p)
return ((n + 1) * p) // (q * r)
print([A341107(n) for n in (0..30)])
A341109
a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240
Offset: 0
- András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
- J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.
- Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.
Cf.
A100655,
A053657 (Minkowski),
A341107,
A341108,
A318256,
A144845,
A163176,
A201637 (Eulerian2),
A036689,
A324370,
A007947,
A324369,
A195441.
-
Epoly := proc(n, x) add(combinat:-eulerian2(n, k)*binomial(x+k, 2*n), k = 0..n) / mul(j-x, j = 1..n): simplify(expand(%)) end:
seq(denom(Epoly(n, x)) / (n!*denom(bernoulli(n, x))), n = 0..30);
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A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k,0,n}],{p, Prime[Range[n]]}];
A144845[n_] := Denominator[Together[BernoulliB[n, x]]];
A163176[n_] := A053657[n] / n!;
Table[(n + 1) A163176[n + 1] / A144845[n], {n, 0, 30}]
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def A341109(n): # uses[A341108, A318256]
return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])
Showing 1-2 of 2 results.
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