A330671 Largest number whose base-n expansion cannot be subdivided to form a sequence of numbers which ordered form a multiple of n+1 when using +, *, and ().

1, 7, 13, 41, 206, 335, 503, 2746, 9898, 13938, 20588, 28390, 38007, 50366, 1006418, 82650, 1865809, 1738855, 2879137, 4024861, 5135433, 5585431, 7932985

Offset

2,2

Comments

Each valid number in a given base n must be a one-digit extension to the right (or left) of another valid number in the same base; otherwise, you could ignore the added digit, make a multiple of n+1 with the rest of the number, then multiply it by the new digit to get a multiple of n+1.

a(n) is always less than n^(n+1).

a(n) also appears to be less than n^(n/2), except at n=3.

Links

Table of n, a(n) for n=2..24.

Tomas Rigaux, Python generator code

Example

For n = 2, the binary notation of a number cannot contain any 0, as you could then construct 0 by multiplying all the digits together, so the only candidates are 1, 11, 111, 1111 (or 1, 3, 7, 15, ... in base 10).
Out of those, if you have at least 2 digits, the number contains the substring '11', which can be multiplied by all the other digits to give 11 (or 3 in base 10), which gives a(2) = 1 as the largest and only solution.
For n = 4, a(n) = 13 can be easily checked using the fact that the base-4 expansion of a valid number cannot contain a 2 and a 3 next to each other, as 2+3 = 5 = n+1.
For n = 10, 12345 is not a valid number as 1+2*3*4*5 = 121 = 11*11.

Prog

(Python) # See Python link

Crossrefs

Sequence in context: A107207 A045464 A134854 * A097444 A241718 A259184

Adjacent sequences: A330668 A330669 A330670 * A330672 A330673 A330674

Keyword

nonn,more,base

Author

Tomas Rigaux, Jan 18 2020

Status

approved