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

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

1,2

Links

Table of n, a(n) for n=1..36.

Example

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.

Mathematica

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 *)

Prog

(Python)
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
(Python)
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

Crossrefs

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

Sequence in context: A321651 A326308 A073901 * A176120 A369595 A220621

Adjacent sequences: A116378 A116379 A116380 * A116382 A116383 A116384

Keyword

base,nonn,more

Author

Robert G. Wilson v, Feb 06 2006

Extensions

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

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

Status

approved