A349771 Integers k with an odd number of digits such that, if m is the number formed from k by deleting its middle digit, then k/m is an integer.

100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253, 260, 264, 275, 280, 286, 297, 300, 315, 330, 341, 352, 360, 363, 374, 385, 390, 396, 400, 405, 440, 451

Offset

1,1

Comments

Generalization of the 7th problem proposed during the 3rd Canadian Mathematical Olympiad in 1971, when this problem asked the question for 5-digit numbers (see Doob reference and link IMO Compendium).

The first 76 terms are the first 76 nontrivial gapful numbers in A108343; then A108343(77) = 1000 while a(77) = 10000.

References

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 7, 1971, page 27, 1993.

Links

Table of n, a(n) for n=1..51.

The IMO Compendium, Problem 7, 3rd Canadian Mathematical Olympiad 1971.

Index to sequences related to Olympiads.

Example

105 / 15 = 7 hence 105 is a term.
290 / 20 = 29/2 hence 290 is not a term.

Mathematica

Select[Range[100, 500], OddQ[(nd = IntegerLength[#])] && Divisible[#, FromDigits[Delete[IntegerDigits[#], (nd + 1)/2]]] &] (* Amiram Eldar, Nov 29 2021 *)

Prog

(Python)
def auptod(maxdigits):
    alst = []
    for d in range(3, maxdigits+1, 2):
        mid = d//2
        for k in range(10**(d-1), 10**d):
            sk = str(k)
            m = int(sk[:mid] + sk[(mid+1):])
            if k%m == 0: alst.append(k)
    return alst
print(auptod(5)) # Michael S. Branicky, Nov 29 2021

Crossrefs

Cf. A108343.

Subsequence of A001633 (odd number of digits).

Sequence in context: A213313 A204584 A108343 * A343552 A204585 A128818

Adjacent sequences: A349768 A349769 A349770 * A349772 A349773 A349774

Keyword

nonn,base

Author

Bernard Schott, Nov 29 2021

Status

approved