A137269 -id:A137269 - cp's OEIS frontend

A133223 Sum of digits of primes (A007605), sorted and with duplicates removed.

2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103

Offset: 1

Lekraj Beedassy, Dec 19 2007

Presumably this is 3 together with numbers greater than 1 and not divisible by 3 (see A001651). - Charles R Greathouse IV, Jul 17 2013. (This is not a theorem because we do not know if, given s > 3 and not a multiple of 3, there is always a prime with digit-sum s. Cf. A067180, A067523. - N. J. A. Sloane, Nov 02 2018)
From Chai Wah Wu, Nov 04 2018: (Start)
Conjecture: for s > 10 and not a multiple of 3, there exists a prime with digit-sum s consisting only of the digits 2 and 3 (cf. A137269). This conjecture has been verified for s <= 2995.
Conjecture: for s > 18 and not a multiple of 3, there exists a prime with digit-sum s consisting only of the digits 3 and 4. This conjecture has been verified for s <= 1345.
Conjecture: for s > 90 and not a multiple of 3, there exists a prime with digit-sum s consisting only of the digits 8 and 9. This conjecture has been verified for s <= 8995.
Conjecture: for 0 < a < b < 10, gcd(a, b) = 1 and ab not a multiple of 10, if s > 90 and s is not a multiple of 3, then there exists a prime with digit-sum s consisting only of the digits a and b. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..5997

Cf. A007605, A067523, A067180, A106754-A106787, A062339, A062341, A062337, A137269.

Corrected by Jeremy Gardiner, Feb 09 2014

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

Original entry on oeis.org

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

A265437 Numbers n such that A265433(n) = 0 and n is not a multiple of 3.

Original entry on oeis.org

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, 1016, 1034, 1040, 1070, 1082, 1133, 1160

Offset: 1

Chai Wah Wu, Dec 10 2015

Conjecture: if n > 4 is in the sequence, then A137269(n) > 0 due to primes with only digits 2, 3, 4.

Chai Wah Wu, Table of n, a(n) for n = 1..100

Cf. A137269, A265433.

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A133223 Sum of digits of primes (A007605), sorted and with duplicates removed.

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

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A265437 Numbers n such that A265433(n) = 0 and n is not a multiple of 3.

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