A365452 Largest number whose digits, in some base, sum to n and include no zeros.

1, 3, 7, 22, 67, 213, 853, 3413, 13653, 66406, 332031, 1660156, 8734003, 52404019, 314424115, 1886544691, 12193514915, 85354604406, 597482230843, 4182375615902, 31414617936457, 251316943491657, 2010535547933257, 16084284383466057, 135502101309790873, 1219518911788117858

Offset

1,2

Comments

There is a proviso with this: the first digit must be one less than the base; otherwise we could claim that a number was in an arbitrarily large base.

The maximum value will always be found by following the initial digit with a string of 1s, which is n candidate values for a(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..597

Formula

a(n) >= A000225(n). - David A. Corneth, Sep 04 2023

a(n) = max_{i=1..n} ((i+1)^(n-i)*(1/i+i)-1/i). - Alois P. Heinz, Sep 09 2023

Example

For n=5, the candidate digits are 11111_2, 2111_3, 311_4, 41_5 and 5_6. These have decimal values 31, 67, 53, 21 and 5, respectively and the largest of is 67, so a(5)=67.

Maple

a:= n-> max((i+1)^(n-i)*(1/i+i)-1/i$i=1..n):
seq(a(n), n=1..25);  # Alois P. Heinz, Sep 06 2023

Prog

(PARI) a(n) = {my(res=2^n-1, v = vector(n, i, 1)); for(i = 2, n, v[i] += v[i-1]; v[i-1] = 0; res = max(res, fromdigits(v, i+1))); res} \\ David A. Corneth, Sep 04 2023
(Python)
def A365452(n): return max(((i + 1)**(n-i)*(i**2 + 1) - 1)//i for i in range(1, n+1)) # Chai Wah Wu, Oct 01 2023

Crossrefs

Cf. A000225.

Sequence in context: A148683 A148684 A148685 * A035353 A242566 A148686

Adjacent sequences: A365449 A365450 A365451 * A365453 A365454 A365455

Keyword

nonn,base

Author

Elliott Line, Sep 04 2023

Extensions

More terms from David A. Corneth, Sep 04 2023

Status

approved