OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 1 for all n > 1.
Conjecture: (ii) For any integer n > 2, the derangement number D(n) given by A000166 has a prime divisor dividing none of those D(k) with 1 < k < n.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..104 (terms 1..81 from Zhi-Wei Sun)
MAPLE
a(4) = 3 since B(4) = 3*5 with 3 dividing none of B(1) = 1, B(2) = 2 and B(3) = 5.
MATHEMATICA
b[n_]:=BellB[n]
f[n_]:=FactorInteger[b[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[b[n]<2, Goto[cc]]; Do[Do[If[Mod[b[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
PROG
(Python)
# Python 3.2 or higher required.
from itertools import accumulate
from sympy import primefactors
A242171_list, bell, blist, b = [1], [1, 1], [1], 1
for _ in range(20):
blist = list(accumulate([b]+blist))
b = blist[-1]
fs = primefactors(b)
for p in fs:
if all([n % p for n in bell]):
A242171_list.append(p)
break
else:
A242171_list.append(1)
bell.append(b) # Chai Wah Wu, Sep 19 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 06 2014
STATUS
approved