This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137269 #46 Jul 21 2024 00:40:45
%S A137269 0,1,1,2,1,0,2,1,0,1,2,0,5,1,0,4,3,0,8,2,0,2,2,0,10,1,0,5,4,0,8,1,0,4,
%T A137269 2,0,17,151,0,7,4,0,13,3,0,812,3,0,17,4,0,12,1,0,13,1,0,6,2,0,18,1,0,
%U A137269 11,1000,0,24,2,0,5,1,0,25,1,0,10,2,0,23,2,0,9,1
%N A137269 Number of primes with maximal digit product for a digit sum of n.
%C A137269 From _Chai Wah Wu_, Dec 04 2015, Nov 03 2018: (Start)
%C A137269 If n > 3 and n == 0 (mod 3) then a(n) = 0 since the digit sum is a multiple of 3.
%C A137269 Primes with digit product maximal among all numbers with the same digit sum (not just maximal among primes) only contain the digits 2, 3 or 4. A digit 0 leads to a digit product 0 which is not maximal. A digit 1 with another digit d (since 1 is not prime, there must be another digit d) can be replaced with the digit d+1 (if d < 9) which preserves the digit sum, but strictly increases the digit product (if d = 9, 1 and 9 can be replaced with 3, 3 and 4 which again increases the digit product). For a digit d > 4, there is a series of digits from the set {2,3,4} whose sum is d and whose product is strictly larger than d. For instance, 5 -> {2,3} whose product is 6. 6 -> {3,3}, 7 -> {3,4}, 8 -> {2,3,3}, 9 -> {3,3,3}. Thus the digit d in a number can be replaced with digits 2, 3, 4 to obtain a number with the same digit sum and a larger digit product. Furthermore, the digits 2 and 4 cannot both appear, the digit 2 cannot appear more than twice and the digit 4 cannot appear more than once since {3,3} also sums to 6 and has product 9 > 8.
%C A137269 This analysis implies the following for n > 3. If n == 1 (mod 3), then primes with maximal digit product among all numbers with the same digit sum (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 among all numbers with the same digit sum (if they exist) have digits 3 and a single digit 2. Values for n for which such primes do not exist are 4, 38, 46, 65, 94, ... In these cases a(n) can still be > 0, but the digit product of these primes is not maximal among all numbers with digit sum n. So far, it seems that in these cases (except for n = 4) these primes also only contain the digits 2, 3, or 4.
%C A137269 (End)
%H A137269 Chai Wah Wu, <a href="/A137269/b137269.txt">Table of n, a(n) for n = 1..802</a>
%e A137269 a(19)=8 and a(20)=2 because we respectively have the 8 primes 333433, 334333, 343333, 2332333, 2333323, 3223333, 3233323, 3332233 all with a maximal digit product of 3^5*2^2 = 972 for a digit sum of 19 and the 2 primes 3233333, 3333233 with maximal digit product 3^6*2 = 1458 for digit sum 20.
%t A137269 Needs["Combinatorica`"]; Table[If[And[n > 3, Divisible[n, 3]], 0, Length@ MaximalBy[Select[FromDigits /@ Flatten[Map[Permutations, Combinatorica`Partitions@ n], 1] /. x_ /; EvenQ@ x -> Nothing, PrimeQ], Times @@ IntegerDigits@ # &]], {n, 24}] (* _Michael De Vlieger_, Dec 11 2015, Version 10 *)
%Y A137269 Cf. A000792, A133223, A136150, A137248, A138975.
%K A137269 nonn,base
%O A137269 1,4
%A A137269 _Lekraj Beedassy_, Apr 05 2008
%E A137269 a(25) and a(28) corrected and a(29)-a(83) added by _Chai Wah Wu_, Nov 30 2015