A110456 Largest prime obtained by concatenation of parts of a distinct partition of n. 0 if no such number exist.

0, 2, 3, 31, 41, 0, 421, 521, 0, 4231, 5231, 0, 7321, 8231, 0, 64231, 74231, 0, 94321, 431021, 0, 754123, 854213, 0, 5431021, 6421013, 0, 8431021, 9431021, 0, 65412103, 75411023, 0, 95421103, 96412103, 0, 643110211, 765324101, 0, 965421013, 975431021, 0, 7542103111, 8543121011

Offset

1,2

Comments

Conjecture a(n) = 0 only for n = 1, or n == 0 (mod 3), n is > 3. Subsidiary sequence (hard): Number of primes generated by concatenation of distinct partitions of n.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..93

Michael S. Branicky, Python program

Example

Distinct partitions of 10 are (10), (9,1), (8,2), (7,3), (6,4), (7,2,1), (6,3,1), (5,3,2), (5,4,1), (4,3,2,1). a(10) = 4231 is the largest prime obtained as a concatenation of (4,2,3,1).

Prog

(PARI) a(n) = my(best=0, cur); forpart(u=n, if(#Set(u)==#u, forperm(u, v, cur=eval(concat(apply(x->Str(x), Vec(v)))); if(isprime(cur), best=max(best, cur))))); best \\ Jason Yuen, Sep 05 2024
(Python) # see link for faster version
from itertools import permutations
from gmpy2 import digits, is_prime, mpz
def dparts(n, high=0): # distinct partitions of n with terms > high
    if n > high: yield (n, )
    for i in range(high+1, n):
        yield from ((i, )+p for p in dparts(n-i, i))
def a(n):
    return max((int(t) for dp in dparts(n) for p in permutations(list(map(str, dp))) if is_prime(t:=mpz("".join(p)))), default=0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 07 2024

Crossrefs

Sequence in context: A272043 A136150 A155056 * A128348 A029973 A344736

Adjacent sequences: A110453 A110454 A110455 * A110457 A110458 A110459

Keyword

base,nonn

Author

Amarnath Murthy, Aug 04 2005

Extensions

Corrected and extended by R. J. Mathar, Feb 08 2008

a(39) and beyond from Michael S. Branicky, Sep 07 2024

Status

approved