A382237 Numbers that are not divisible by the sum of any subset of their digits.

23, 29, 34, 37, 38, 43, 46, 47, 49, 53, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 223, 227, 229, 233, 239, 249, 253, 257, 263, 267, 269, 277, 283, 293, 299, 307, 323, 329, 334, 337, 338, 346, 347, 349, 353, 356, 358, 359, 367, 373, 376, 377, 379, 380, 383, 386, 388, 389, 394, 397, 398, 403

Offset: 1

Author

Sergio Pimentel, Mar 19 2025

Keywords

• nonn
• base

Comments

This sequence has density zero since no numbers with the digit '1' are in it. The sequence is infinite. Example: Numbers like 23, 203, 2003, 20003, etc. are included because none of them is divisible by 2, 3, or 5.

Conjecture: after a sufficiently large n this sequence grows faster than the prime numbers.

Examples

			358 is in the sequence because it can't be divided by 3, 5, 8, (3+5)=8, (3+8)=11, (5+8)=13 or (3+5+8)=16.
289 is not in the sequence because it can be divided by (8+9)=17.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A082943, A038772, A005349, A065877, A228017, A382239.

Programs

Maple

filter:= proc(n) local L,S;
  L:= convert(n,base,10);
  andmap(s -> s=0 or n mod s <> 0, map(convert,combinat:-choose(L),`+`))
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2025

PARI

isok(k) = my(d=digits(k)); forsubset(#d, s, my(ss=sum(i=1, #s, d[s[i]])); if (ss && !(k % sum(i=1, #s, d[s[i]])), return(0))); return(1); \\ Michel Marcus, Mar 27 2025

Python

from itertools import chain, combinations
def powerset(s): # skipping empty set
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
def ok(n): return all(n%t!=0 for s in powerset(list(map(int, str(n)))) if (t:=sum(s))>0)
print([k for k in range(1, 404) if ok(k)]) # Michael S. Branicky, Apr 01 2025