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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 (list; graph; refs; listen; history; text; internal format)
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 A091265 A262373

Adjacent sequences:  A242168 A242169 A242170 * A242172 A242173 A242174

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 06 2014

STATUS

approved

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Last modified August 2 14:18 EDT 2021. Contains 346428 sequences. (Running on oeis4.)