A113625
Irregular triangle in which the n-th row contains all primes having digit sum n (not containing the digit '0') in increasing order.
Original entry on oeis.org
2, 11, 3, 13, 31, 211, 5, 23, 41, 113, 131, 311, 2111, 7, 43, 61, 151, 223, 241, 313, 331, 421, 1123, 1213, 1231, 1321, 2113, 2131, 2221, 2311, 3121, 4111, 11113, 11131, 11311, 12211, 21121, 21211, 22111, 111121, 111211, 112111, 17, 53, 71, 233, 251, 431, 521
Offset: 2
Starting with row 2, the table is
2, 11
3
13, 31, 211
5, 23, 41, 113, 131, 311, 2111
none
7, 43, 61, 151, 223, 241, 313, 331, 421, 1123,...
Cf.
A110741 (with contraints on number of digits).
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with(combinat):
b:= proc(n, i, l) option remember; `if`(n=0, select(isprime,
map(x-> parse(cat(x[])), permute(l))), `if`(i<1, [],
[seq(b(n-i*j, i-1, [l[],i$j])[], j=0..n/i)]))
end:
T:= n-> sort(b(n, 9, []))[]:
seq(T(n), n=2..8); # Alois P. Heinz, May 25 2013
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Table[If[n > 3 && Mod[n, 3] == 0, {}, p = IntegerPartitions[n]; u = {}; Do[t = Permutations[i]; u = Union[u, Select[FromDigits /@ t, PrimeQ]], {i, p}]; u], {n, 2, 14}]
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
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.
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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 *)
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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
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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
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