A330083 a(n) is the smallest number k > 0 such that for each b = 2..n the base-b expansion of k has exactly n - b zeros.

1, 2, 10, 18, 271

Offset

2,2

Comments

This list is complete. Proof: When converting base 2 to base 4, we can group the digits in base 2 into pairs from the least significant bit. We then convert pairs into single digits in base 4 as 00 -> 0, 01 -> 1, 10 -> 2, 11 -> 3. This always causes the number of zeros to go to half or less than half. For all n >= 7, n-4 is greater than (n-2)/2, so the condition is impossible. - Christopher Cormier, Dec 08 2019

Does k exist for every n >= 2?

a(7) > 10^7, if it exists.

a(7) > 2^64, if it exists. - Giovanni Resta, Dec 01 2019

Links

Table of n, a(n) for n=2..6.

Example

For n = 6: The base-b expansions of 271 for b = 2..6 are shown in the following table:
b | base-b expansion | number of zeros
---------------------------------------
2 |        100001111 | 4
3 |           101001 | 3
4 |            10033 | 2
5 |             2041 | 1
6 |             1131 | 0

Prog

(PARI) count_zeros(vec) = #setintersect(vecsort(vec), vector(#vec))
a(n) = for(k=1, oo, for(b=2, n, if(count_zeros(digits(k, b))!=n-b, break, if(b==n, return(k)))))

Crossrefs

Sequence in context: A254059 A346551 A180591 * A322951 A156446 A032685

Adjacent sequences: A330080 A330081 A330082 * A330084 A330085 A330086

Keyword

nonn,base,fini,full

Author

Felix Fröhlich, Dec 01 2019

Extensions

Value of a(2) adjusted by Felix Fröhlich, Dec 09 2019

Status

approved