A069870 -id:A069870 - cp's OEIS frontend

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

Amarnath Murthy, Apr 21 2002

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

			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.

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

Cf. A069870, A004022.

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

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]

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

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

A116381 Number of compositions of n which are prime when concatenated and read as a decimal string.

Original entry on oeis.org

0, 2, 1, 3, 7, 0, 29, 27, 0, 90, 236, 0, 758, 1039, 0, 3949, 9325, 0, 32907, 51243, 0, 184458, 426372, 0, 1552101, 2537233, 0, 9526385, 21117111, 0, 78112040, 134568638, 0, 505079269, 1096046406, 0

Offset: 1

Robert G. Wilson v, Feb 06 2006

			The eight compositions of 4 are 4,13,31,22,112,121,211,1111 of which 3 {13,31,211} are primes.
Primes for n=11 are: 11, 29, 47, 83, 101, 137, 173, 191, 227, 263, 281, 317, 353, 443, 461, 641, 821, 911, 1163, 1181, ..., 131111111, 212111111, 1111111121, 1111211111, 1121111111.

Cf. A069869, A069870; not the same as A073901.

f[n_] := If[n > 5 && Mod[n, 3] == 0, 0, Block[{len = PartitionsP@ n, p = IntegerPartitions[n], c = 0}, Do[c = c + Length@ Select[ FromDigits /@ Join @@@ IntegerDigits /@ Permutations@ p[[i]], PrimeQ@# &], {i, len}]; c]]; Array[f, 28] (* Robert G. Wilson v, Aug 03 2012 *)

from sympy import isprime
from sympy.utilities.iterables import partitions, multiset_permutations
def a(n):
    c = 0
    for p in partitions(n):
        plst = [k for k in p for _ in range(p[k])]
        s = sum(sum(map(int, str(pi))) for pi in plst)
        if s != 3 and s%3 == 0: continue
        for m in multiset_permutations(plst):
            if isprime(int("".join(map(str, m)))):
                c += 1
    return c
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Nov 19 2022

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
from sympy import isprime
def A116381(n): return sum(1 for p in partitions(n) for a in multiset_permutations(Counter(p).elements()) if isprime(int(''.join(str(d) for d in a)))) if n==3 or n%3 else 0 # Chai Wah Wu, Feb 21 2024

a(29)-a(33) from Michael S. Branicky, Nov 19 2022

a(34)-a(36) from Michael S. Branicky, Jul 10 2023

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

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A116381 Number of compositions of n which are prime when concatenated and read as a decimal string.

Views

Author

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Examples

Crossrefs

Programs

Mathematica

Python

Python

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