A348730 Largest remainder of an n-digit number when divided by its sum of digits.

0, 15, 24, 31, 43, 49, 58, 69, 79, 86, 95, 103, 114, 124, 130, 142, 150, 159, 169, 177, 185, 195, 203, 214, 223, 231, 241, 249, 259, 267, 275, 286, 295, 303, 312, 321, 330, 340, 349, 357, 367, 375, 383, 394, 403, 411, 421, 429, 439, 448, 456, 465, 475, 482, 493, 501

Offset

1,2

Comments

When dividing a number by N, the remainder can be at most N - 1. The largest sum of digits that an n-digit number can reach is 9*n. According to this, a(n) < 9*n is always the case.

Is the sequence strictly increasing? a(n) is achieved by a single n-digit number when n = 2, 3, 5, 19, 24, 27, 32, 38, ... (Cf. A348900) - Chai Wah Wu, Nov 02 2021

Links

David A. Corneth, Table of n, a(n) for n = 1..1780 (first 820 terms from Chai Wah Wu)

David A. Corneth, PARI program

Heinrich Hemme, Was ist der größtmögliche Rest? Hemmes mathematische Rätsel, Oct 28 2021.

Volker Wagner, Kleine Rechnerei, Knobelforum, Jan 24 2020.

Example

79 = 15 mod 16
799 = 24 mod 25
9599 = 31 mod 32
98999 = 43 mod 44
599999 = 49 mod 50
8999998 = 58 mod 61
etc.

Maple

sod:=proc(n) # sum of digits
  local N;
  N:=convert(n, base, 10);
  add(i, i=N);
end;
a:=proc(k)local n, i;
  for n from 1 to k do
    maxR||n:=0: # largest remainder
    for i from 10^(n-1) to 10^n-1 do
      if i mod sod(i)>maxR||n then
         maxR||n:=i mod sod(i);
       fi;
    od:
  od:
  return(seq(maxR||n, n=1..k));
end;

Mathematica

a[n_] := Max[Mod[#, (Plus @@ IntegerDigits[#])] & /@ Range[10^(n - 1), 10^n - 1]]; Array[a, 7] (* Amiram Eldar, Oct 31 2021 *)

Prog

(Python)
def sd(n): return sum(map(int, str(n)))
def a(n): return max(n%sd(n) for n in range(10**(n-1)+1, 10**n))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Oct 31 2021
(PARI) See Corneth link
(Python)
from functools import lru_cache
from sympy.combinatorics.partitions import IntegerPartition
from sympy.utilities.iterables import partitions, multiset_permutations
@lru_cache(maxsize=None)
def intpartition(n, m): return tuple(''.join(str(d) for d in IntegerPartition(p).partition+[0]*(m-s)) for s, p in partitions(n, k=9, m=m, size=True))
def A348730(n):
    l, c, k = 9*n, 0, 10**(n-1)
    while l-1 > c:
        c = max(c, max(s % l for s in (int(''.join(q)) for p in intpartition(l, n) for q in multiset_permutations(p)) if s >= k))
        l -= 1
    return c # Chai Wah Wu, Nov 03-08 2021

Crossrefs

Cf. A070635, A348900.

Sequence in context: A269314 A269316 A081829 * A079721 A362812 A089952

Adjacent sequences: A348727 A348728 A348729 * A348731 A348732 A348733

Keyword

nonn,base

Author

Martin Renner, Oct 31 2021

Extensions

a(8)-a(10) from Michael S. Branicky, Oct 31 2021

More terms from David A. Corneth, Oct 31 2021

Status

approved