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URL: https://oeis.org/A265433

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A265433
Number of primes with digit sum n whose digit product is maximal among all numbers with digit sum n.
2
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
1,7
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(Python)
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
CROSSREFS
Sequence in context: A036581 A369462 A135055 * A298247 A035148 A155077
KEYWORD
nonn,base
AUTHOR
Chai Wah Wu, Dec 08 2015
STATUS
approved