A265437 -id:A265437 - cp's OEIS frontend

A265433 Number of primes with digit sum n whose digit product is maximal among all numbers with digit sum n.

0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 2, 0, 5, 1, 0, 4, 3, 0, 8, 2, 0, 2, 2, 0, 10, 1, 0, 5, 4, 0, 8, 1, 0, 4, 2, 0, 17, 0, 0, 7, 4, 0, 13, 3, 0, 0, 3, 0, 17, 4, 0, 12, 1, 0, 13, 1, 0, 6, 2, 0, 18, 1, 0, 11, 0, 0, 24, 2, 0, 5, 1, 0, 25, 1, 0, 10, 2, 0, 23, 2, 0, 9, 1

Offset: 1

Chai Wah Wu, Dec 08 2015

If n == 0 mod 3, then a(n) = 0.
If n == 1 mod 3, then primes with maximal digit product (if they exist) have digits 3 and either two digits 2 or a single digit 4.
If n == 2 mod 3, then primes with maximal digit product (if they exist) have digits 3 and a single digit 2 (see comment in A137269).
If n == 0 mod 3 or a(n) > 0, then a(n) = A137269(n). Terms a(n) coincide with A137269 except for n = 4, 38, 46, 65, 94, 107, 116, 128, 131, 140, 143, 149, 152, 170, 188, 227, ..., 767 (and most likely other n > 767). For these values of n, a(n) = 0 and A137269(n) > 0.
Conjecture: For n > 4, if n <> 0 mod 3 and a(n) = 0, then A137269(n) > 0 due to primes with only digits 2, 3, or 4.

			See examples in A137269. a(4) = 0 since the maximal digit product is 4 corresponding to the numbers 22 and 4, neither of which is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..2001

Cf. A137269, A265437.

f[n_] := Block[{g, a265437 = {1, 4, 38, 46, 65, 94, 107, 116, 128, 131, 140, 143, 149, 152, 170, 188, 227, 230, 248, 272, 293, 302, 317, 335, 344, 359, 371, 382, 404, 488, 530, 533, 551, 584, 626, 647, 722, 767, 803, 815, 866, 875, 893, 914, 920}},
  g[k_] := Length@ MaximalBy[k, Times @@ IntegerDigits@ # &];
  Which[MemberQ[a265437, n], 0,
   1 < n <= 3, 1,
   Mod[n, 3] == 0, 0,
   Mod[n, 3] == 1, g@ Select[FromDigits /@ Apply[Join, Map[Permutations, {Join[Table[3, {Floor[n/3] - 1}], {2, 2}], Join[Table[3, {Floor[n/3] - 1}], {4}]}]] /. x_ /; EvenQ@ x -> Nothing, PrimeQ],
   Mod[n, 3] == 2, g@ Select[FromDigits /@ Permutations@ Join[Table[3, {Floor[n/3]}], {2}] /. x_ /; EvenQ@ x -> Nothing, PrimeQ],
True, -1]] (* Michael De Vlieger, Dec 11 2015, Version 10, reliant on values of A265437 *)

from _future_ import division
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime
def A265433(n):
    if n == 1:
        return 0
    if n == 3:
        return 1
    if (n % 3) == 0:
        return 0
    else:
        pmaxlist = ['3'*(n//3) + '2'] if (n % 3 == 2) else ['3'*(n//3 -1) + '22','3'*(n//3 -1) + '4']
        return sum(1 for p in pmaxlist for k in multiset_permutations(p) if isprime(int(''.join(k)))) # Chai Wah Wu, Dec 11 2015

cp's OEIS Frontend

A265433 Number of primes with digit sum n whose digit product is maximal among all numbers with digit sum n.

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