A178157 -id:A178157 - cp's OEIS frontend

A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 150, 200, 210, 220, 240, 250, 300, 330, 360, 400, 420, 440, 480, 500, 510, 520, 550, 600, 630, 660, 700, 770, 800, 840, 880

Offset: 1

Rémy Sigrist, May 01 2025

This sequence is infinite as ten times a term is also a term.
All terms are of the form A037124(k) or A037124(k) + d where k > 0 and d divides A037124(k) while having strictly less decimal digits as A037124(k).
Empirically, all terms have either one or two nonzero decimal digits.

			The number 240 is divisible by 2, 24, 240, 4 and 40, so 240 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10314
Rémy Sigrist, PARI program

Cf. A037124, A078546, A175381 (binary variant), A178157, A218978.

Select[Range[880],AllTrue[#/Select[FromDigits/@Subsequences[IntegerDigits[#]],#>0&],IntegerQ]&] (* James C. McMahon, May 13 2025 *)

is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if (n % fromdigits(d[i..j], base),
                    return (0);););););
    return (1); }

PARI
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def ok(n):
    s = str(n)
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and all(n%v == 0 for ss in subs if (v:=int(ss)) > 0)
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, May 09 2025

A308237 Numbers m not ending with 0 that contain a digit, other than the leftmost digit, that can be removed such that the resulting number d divides m.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99, 105, 108, 121, 132, 135, 143, 154, 165, 176, 187, 192, 195, 198, 225, 231, 242, 253, 264, 275, 286, 297, 315, 341, 352, 363, 374, 385, 396, 405, 451, 462, 473, 484, 495, 561, 572, 583, 594, 671, 682, 693

Offset: 1

Bernard Schott, May 16 2019

When m is a term, then, necessarily, the digit that is removed is the second from the left.
This sequence is finite with 95 integers and the greatest term is 180625. The number of terms with respectively 2, 3, 4, 5, 6 digits is 23, 44, 10, 17, 1.
The obtained quotients m/d belong to: { 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19 } (all proofs in Diophante link).

			264 is a term because 264/24 = 11.
34875 is a term because 34875/3875 = 9.

Marius A. Burtea, Table of n, a(n) for n = 1..95
Diophante, A333. Un chiffre à la trappe, Oct. 2011 (in French).

Cf. A034837, A178157.

m=1;
for u=10:700 digit=dec2base(u,10)-'0';
   if digit(length(digit))~=0 aa=str2num(strrep(num2str(digit), ' ', ''));
      digit(2)=[]; a=str2num(strrep(num2str(digit), ' ', ''));
if mod(aa,a)==0 sol(m)=u;  m=m+1;  end; end; end;
sol % Marius A. Burtea, May 16 2019

Select[Range[700], With[{m = #}, And[Mod[#, 10] != 0, AnyTrue[FromDigits@ Delete[IntegerDigits[m], #] & /@ Range[2, IntegerLength@ m], Mod[m, #] == 0 &]]] &] (* Michael De Vlieger, Jun 09 2019 *)

isok(m) = {if (m % 10, my(d=digits(m)); for (k=2, #d, mk = fromdigits(vector(#d-1, i, if (iMichel Marcus, Jun 21 2019

A359841 Integers Xd which are divisible by X, where d is the last decimal digit.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420

Offset: 1

Bernard Schott, Jan 15 2023

Integers k such that k is divisible by A059995(k).

This sequence consists of {the thirty-two 2-digit terms of A034837 (from 10 up to 99)} Union {the positive multiples of 10 (A008592\{0})}.

N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, problem 15, pp. 11 and 102, Dover publications, Inc., New York, 1993.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A034837, A059995, A178157, A292683 (similar but with dX).

Subsequence: A008592\{0}.

Select[Range[10, 500], Divisible[#, Floor[#/10]] &] (* Amiram Eldar, Jan 15 2023 *)

isok(k) = (k>9) && (k % (k \ 10) == 0); \\ Michel Marcus, Jan 20 2023

def ok(n): return n > 9 and n%(n//10) == 0
print([k for k in range(421) if ok(k)]) # Michael S. Branicky, Jan 15 2023

def A359841(n): return (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99)[n-1] if n <= 32 else (n-23)*10 # Chai Wah Wu, Jan 20 2023

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A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

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A308237 Numbers m not ending with 0 that contain a digit, other than the leftmost digit, that can be removed such that the resulting number d divides m.

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Examples

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MATLAB

Mathematica

PARI

A359841 Integers Xd which are divisible by X, where d is the last decimal digit.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python