A069869 Largest prime that is a concatenation of the parts of a partition of n, or 0 if no such prime exists.

0, 11, 3, 211, 2111, 0, 112111, 1111211, 0, 11131111, 1121111111, 0, 111211111111, 2111111111111, 0, 31111111111111, 212111111111111, 0, 1111111111111111111, 2111111111111111111, 0, 111111111111111121111, 11111111111111111111111, 0, 211111111111111111111111

Offset

1,2

Comments

Conjecture: a(n) = 0 only for n = 1 or n = 3k with k>1.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..200

Example

a(4) = 211 as the partitions of 4 are (4), (3,1), (2,2), (2,1,1) (1,1,1,1). The primes that can be formed are 13, 31, 211 and 211 is the largest prime using a partition.

Maple

with(combinat):
a:= proc(n) local k, w;
      if n=1 or n>3 and irem(n, 3)=0 then return 0 fi;
      for k from 0 do w:= max(select(isprime,
        map(x-> parse(cat(x[])), [seq(permute(i)[],
          i=map(x->[x[], 1$(n-k)], partition(k)))]))[]);
        if w>0 then return w fi
      od
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, May 25 2013

Mathematica

f[n_] := If[ PrimeQ@n, n, If[n > 5 && Mod[n, 3] == 0, 0, Block[{len = PartitionsP[n], p = IntegerPartitions[n], t = {}}, Do[ AppendTo[t, Select[FromDigits /@ Join @@@ IntegerDigits /@ Permutations@p[[i]], PrimeQ@# &]], {i, len}]; t = Union@Flatten@t; If[Length@t > 0, Max@t, 0]] ]]; Array[f, 29]

Prog

(Python)
from collections import Counter
from operator import itemgetter
from sympy.utilities.iterables import partitions, multiset_permutations
from sympy import isprime
def A069869(n):
    smax, m = 0, 0
    if n==3 or n%3:
        for s, p in sorted(partitions(n, size=True), key=itemgetter(0), reverse=True):
            if s<smax:
                break
            for a in multiset_permutations(Counter(p).elements()):
                if isprime(k:=int(''.join(str(d) for d in a))):
                    m = max(k, m)
            if m>0:
                smax=s
    return m # Chai Wah Wu, Feb 21 2024

Crossrefs

Cf. A069870, A004022.

Sequence in context: A157883 A038317 A157781 * A079841 A110782 A088073

Adjacent sequences: A069866 A069867 A069868 * A069870 A069871 A069872

Keyword

nonn,base

Author

Amarnath Murthy, Apr 21 2002

Extensions

More terms from David Wasserman, Apr 30 2003

a(8) corrected and a(16)-a(24) added by Robert G. Wilson v, Feb 06 2006

a(25) from Alois P. Heinz, May 25 2013

Status

approved