A065851(n) is the maximum number of n-digit primes which can be made by permuting n digits, and a(n) is the smallest number which reaches this maximum.

a(n) ishas theits relevant digits sorted and not beginning with 0, and may or may not be one of the primes (it is for n = 1 to 7, but not at n = 8).

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c[smallest("".["".join(sorted(str(p))))] += )))] += 1

m = min(c.most_common(1))), key=lambda x:smallest(x[0]))

return smallest(m[0] # ]) # m[1] generates A065851

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Michael S. Branicky: slightly faster to do at end

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c["".[smallest("".join(sorted(str(p)))] += ))))] += 1

return smallest(m[0]) # ] # m[1] generates A065851

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Michael S. Branicky: Yes.  Updated.

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Kevin Ryde: Oh, also it's possible the maximum count is attained by more than one digit set, eg. n=3 of the example.  In that case pick the smallest of the "smallest()".  I think it doesn't happens for n=5..11, but have to watch.

print([a(n) for n in range(21, 78)]) # Michael S. Branicky, May 28 2024

Michael S. Branicky: Indeed, Kevin!  Shorter too!

from sympy import isprimenextprime

from itertoolscollections import combinations_with_replacementCounter

from sympy.utilities.iterables import multiset_permutations

s = "".join(nzt) if "0" not in t else nz[0]+"0"*t.count("0")+nz[1:]

mxc, argmx = -p = Counter(), nextprime(10**(n-1, None))

for u in combinations_with_replacement("0123456789", n):

s, first, ufirst = 0, False, None

if sum(int(ui) for ui in u)%3 != 0:

while p < 10**n:

c["".join(sorted(str(p)))] += 1

p = nextprime(p)

m = min(c.most_common(1))

return smallest(m[0]) # m[1] generates A065851

print([a(n) for pn in multiset_permutationsrange(u):2, 7)]) # _Michael S. Branicky_, May 28 2024

if p[0] != "0" and isprime(int("".join(p))):

s += 1

if not first:

ufirst, first = int("".join(p)), True

if s > mx:

mx, argmx = s, u # use ufirst for smallest prime

return smallest(argmx) # mx here generates A065851

print([a(n) for n in range(1, 8)]) # Michael S. Branicky, May 27 2024

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Gonzalo Martínez: Thank you very much, Kevin and Michael :D

Kevin Ryde: On the code front, I tried the other direction: Go through all primes, get their digit sets (multi-sets), count how many of each.  There's a lot of primes, but relatively few resulting sets.  Eg. n=12 is binomial(12+9,9) = about 300k.

2, 13, 149, 1237, 13789, 123479, 1235789, 12345679, 102345679, 1123456789, 10123456789

a(9)-a(1011) from Michael S. Branicky, May 27 2024

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