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.

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

Cf. A000040, A000110, A000166, A242169, A242170, A242173.

Sequence in context: A176914 A194010 A229609 * A254790 A353717 A091265

Adjacent sequences: A242168 A242169 A242170 * A242172 A242173 A242174

Keyword

nonn

Author

Zhi-Wei Sun, May 06 2014

Status

approved