A169858 Smallest integer k such that k or one of its left substrings (or prefixes, regarded as an integer) is divisible by any integer from {1,2,...,n}.

1, 2, 6, 12, 45, 60, 245, 245, 504, 504, 5049, 5049, 10296, 11760, 11760, 11760, 56160, 56160, 198016, 198016, 1008159, 1323008, 2340849, 6240366, 13442580, 13442580, 37536408, 37536408, 75432065, 75432065, 180092645, 319800096, 319800096, 800640126, 2201169600, 2201169600, 3780487275, 5250966084, 5250966084, 6832425609, 36960308625, 36960308625, 62244072512, 62244072512, 62244072512, 62244072512, 372960042489, 372960042489

Offset: 1

Author

Hugo van der Sanden, Jun 01 2010

Keywords

• nonn
• base

Examples

			a(5) = 45 as the left substrings of 45 are {4, 45} and for every d in {1,2,...,n} = {1, 2, 3, 4, 5} there is a left substring of 45 such that d | 45. That is: 1 | 4, 2 | 4, 3 | 45, 4 | 4, 5 | 45. - _David A. Corneth_, Jun 09 2023
		

Links

• Hugo van der Sanden A169858: C source code to calculate terms.

Crossrefs

Cf. A177834, A175516, A003418.

Programs

Python

from itertools import count, islice
def agen(): # generator of terms
    n = 1
    for k in count(1):
        s = str(k)
        prefixes = [int(s[:i+1]) for i in range(len(s))]
        if all(any(ki%m == 0 for ki in prefixes) for m in range(1, n+1)):
            yield k; n += 1
            while any(ki%n == 0 for ki in prefixes):
                yield k; n += 1
print(list(islice(agen(), 20))) # Michael S. Branicky, Jun 09 2023

Formula

a(n) = min m: forall d in {1..n}: exists k in {0..log_10(m)}: d | floor(m / 10^k).

a(n) <= A003418(n). - Michael S. Branicky, Jun 09 2023

Extensions

Corrected and extended by Hugo van der Sanden, Jun 04 2010 (errors reported by Zak Seidov).