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A242171
Least prime divisor of B(n) which does not divide any B(k) with k < n, or 1 if such a primitive prime divisor of B(n) does not exist, where B(n) is the n-th Bell number given by A000110.
11
1, 2, 5, 3, 13, 7, 877, 23, 19, 4639, 22619, 37, 27644437, 1800937, 251, 241, 255755771, 19463, 271, 61, 24709, 17, 89, 123419, 367, 101, 157, 67, 75979, 107, 11, 179167, 5694673, 111509, 980424262253, 193, 44101, 5399, 6353, 3221
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