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

Author

Amarnath Murthy, Apr 21 2002

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A069870, A004022.

Programs

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]

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 s0:
                smax=s
    return m # Chai Wah Wu, Feb 21 2024

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